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spectral weight in the core regions of the Mn are much higher than in bulk. As an example, the upper accep- tor for the 〈 110 〉 R 0 . 71 a has around 50% spectral weight on the As in-between the Mn, to be compared with 17% in bulk, which is still high relative other bulk configurations. Wave functions for other surface pairs have a much higher 

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maximum spectral weight than in bulk, with values in the typical range 20%-30%. Overall, the spectral weight maximum for a given pair in bulk is approximately an order of magnitude smaller than the corresponding pair at the surface, with bulk wave functions that are much more spread out in the lattice. 

As shown in the first two rows of Fig. 22, some bonding/antibonding characteristics in the acceptor wave functions can be seen for the closest pairs 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a,}$$_{1}$$_{.}$$_{41}$$_{a}$ in the FM configuration, which exhibit magnetic quasi-easy planes. Both acceptor wave func- tions for the 〈 110 〉$_{d}$$_{=2}$$_{.}$$_{12}$$_{a}$ Mn pair exhibit a hint of an- tibonding character. Note that this pair is characterized by the magnetic anisotropy landscape of a single Mn im- purity. 

For 〈 100 〉$_{d}$$_{=}$$_{a}$ each acceptor level is occupying both Mn sites, but a separation occurs for 〈 100 〉$_{d}$$_{=2}$$_{a,d}$$_{=3}$$_{a}$ such that each acceptor has more pronounced LDOS on one of the sites. The same association of acceptors spatially bound to one site, appears for the other directions - a pattern indicative of weak hybridization. Finally the 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{22}$$_{a}$ has some hint of bonding/antibonding pattern, but not as pronounced as for bulk. 

In the LDOS for the AFM configurations (see Fig. 23), the 〈 110 〉 pairs show no spatial separation between the acceptors and no bonding/antibonding behavior. The 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{22}$$_{a}$ has upper and lower acceptor wave functions that are localized to the separate Mn sites, whereas the rest of the pairs are similar to the bulk counterparts (see Fig. 12), but with a more localized LDOS signature. 

## 3. Acceptor splitting vs. effective exchange constant J 

Finally, we have calculated the effective exchange con- stant for the Mn pairs in the surface, which we compare 

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