19 fects not included in our tight-binding approach could partly be responsible for the larger splittings. Analysis of the LDOS for the two acceptor states shows that these are mainly concentrated in the surroundings of the Mn atoms; for intermediate separations, when the splitting is sizable, the higher state and lower state are of the bond- ing and anti-bonding type respectively. These results are all consistent with experiment, 4,5 indicating that the TB model correctly describes the electronic properties of the coupled acceptor states in GaAs associated with the magnetic impurities. An important question that we have addressed here is whether or not the acceptor splitting, which is acces- sible by STM measurements, is directly related to the effective exchange coupling between the Mn local mo- ments. The present work demonstrates that, at least within the model considered here, this relationship is not very sharp: although both quantities decrease with sep- aration and are typically anisotropic, the acceptor split- ting displays a non-monotonic behavior for a few pair orientations which is not found in the exchange coupling. In our model, the dependence of the latter on Mn sepa- ration and pair orientation is better represented by the overall bandwidth 50 of the six impurity levels present in the gap (see Fig. 17). In particular, Our calculations demonstrate that exchange interactions between two Mn ions in (Ga,Mn)As are closely related to the splitting be- tween the two acceptor levels that are occupied by holes and the four lower energy acceptor levels that are occu- pied by electrons. This splitting is not readily measured by STM experiments which are strongly influenced by Coulomb interaction energies when the number of holes bound to the acceptor complex is increased above two by removing electrons from the system. It is possible that infrared spectroscopy of transitions between iso-charge levels of the acceptor complex could be more effective. These results support the idea that the stability and strength of the ferromagnetic coupling involves the hy- bridization of all impurity levels and not only the two topmost ones occupied by holes. Our estimate of the exchange coupling is in qualitative agreement with re- sults obtained from ab initio calculations, with a few no- ticeable discrepancies analyzed in detailed in Sec. III A 4, which are most likely due to the limitations of both meth- ods and deserve further investigation. When the two Mn atoms are inserted in the (110) sur- face as in experiment, we typically find much deeper This work was supported by the Welch Foundation and the National Science Foundation under grant DMR- 0606489, the Faculty of Natural Sciences at Kalmar Uni- versity, the Swedish Research Council under Grant No: 621-2004-4439, and by the Office of Naval Research un- der grant N00014-02-1-0813. We would like to thank J. Gupta, A. Yazdani, P. M. Koenraad, J. K. Garleff, A. P. Wijnheijmer and C. F. Hirjibehedin for useful dis- cussions. acceptor states with strongly localized wave functions around the impurities. Weaker hybridization causes the acceptor splitting to decay more rapidly with separation, although in a manner qualitatively similar to bulk. With the exception of the most closely spaced Mn pair, the surface acceptor splittings seem to agree less well with experiment than our bulk results. The exchange con- stant is likewise very small except at the shortest sepa- rations and can even change sign, favoring an antiferro- magnetic coupling. These results are a clear indication that our semiphenomenological microscopic model, de- rived mainly from bulk (Mn,Ga)As properties, is not able to quantitatively capture the complexity of the surface states. This is an area that could benefit from additional work. The model investigated here makes predictions for the spin-orbit induced magnetic anisotropy for pairs of Mn atoms in GaAs. Some of these predictions could be tested in STM experiments by applying an external field. When the Mn pair is in a bulk environment, there is a uniax- ial magnetic anisotropy along the axis of the pair with anisotropy barriers of 10-20 meV. When the pair is in a (110) surface, the behavior of the anisotropy energy land- scape is for most pairs similar in character and magnitude to that of a single Mn atom in the (110) surface. $^{10}$. In the present approach the local moment degrees of freedom have been treated as classical variables. The study of the quantum spin dynamics of two interacting Mn in GaAs is an important and interesting subject both theoretically and in view of future experiments. Work in this direction, based on Berry-phase quantization of the semi-classical local moments considered here, is under- way. V. ACKNOWLEDGMENTS 1 A. M. Yakunin, A. Y. Silov, P. M. Koenraad, W. V. Roy, J. D. Boeck, J.-M. Tang, and M. E. Flatt´e, Phys. Rev. Lett 92 , 216806 (2004). 2 A. M. Yakunin, A. Y. Silov, P. M. Koenraad, J.-M. Tang, M. E. Flatt´e, W. V. Roy, J. D. Boeck, and J. H. Wolter, Phys. Rev. Lett 95 , 256402 (2005). 3 D. Kitchen, A. Richardella, and A. Yazdani, J. Supercond. 18 , 23 (2005). 4 D. Kitchen, A. Richardella, J.-M. Tang, M. E. Flatt´e, and A. Yazdani, Nature 442 , 436 (2006). 5 D. Kitchen, A. Richardella, P. Roushan, J.-M. Tang, M. E. Flatt´e, and A. Yazdani, Jour. App. Phys. 101 , 515 (2007). 6 J. K. 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