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We first consider the 〈 110 〉 pairs in the FM configu- ration (see Fig. 11). For the special case where the Mn atoms are separated by a common nearest neighbor As, the results change significantly depending on whether the off-site Coulomb correction is additive ( 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ ) or non-additive ( 〈 110 〉 ∗ d =0 . 71 $_{a}$). We see that the effect of a non-additive Coulomb correction on the common As is to delocalize the acceptor wave functions. A more delocal- ized wave function generates larger anisotropy energies, as shown in Fig. 9. At the shortest Mn separation, both acceptor wave functions have bonding character , with the maximum spectral weight on the common As neighbor located between them. This situation is captured by the toy model, where the large hybridization occurring at small Mn separation gives rise to the level structure shown in Figs. 5 and 6, with both acceptors being of the bonding type. As the Mn ions move apart along 〈 110 〉 , the lower acceptor state develops more antibonding char- acter with a significant decrease of the spectral weight between the two Mn sites, whereas the upper acceptor re- mains in a bonding-like state. Within our toy model, this implies that the two acceptor states correspond to bond- ing and antibonding states arising from the hybridization of degenerate levels in energy, spin and orbital character, as shown in Fig. 4. 

In agreement with our results, the STM experiments for the 〈 110 〉 pair find clear evidence that the two Mn- induced acceptor states have bonding and antibonding character, with the bonding state occurring at higher energies. 4,5 An antibonding character for the lower en- ergy state is observed experimentally also for the Mn pair with the shortest separation. 5 While this does not seem to be case in our bulk 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ calculations (both states being essentially “bonding”), we do find that the value of the maximum LDOS on the As in between the Mn’s is around twice as large for the upper acceptor wave function. 

For the 〈 100 〉 pairs the upper acceptor exhibits a more localized signature, whereas the lower acceptor is more extended because it is closer to the valence continuum. Bonding and antibonding formations cannot be clearly seen, indicating that the hybridization between the two holes is much weaker along the 〈 100 〉 . This conclusion is also supported by the very small acceptor-energy split- ting, as shown in Fig. 21. In the remaining directions 〈 211 〉 and 〈 111 〉 (rows 8-11), bonding and antibonding characters are visible, with more and less spectral weight in the region between the Mn, respectively. 

In the AFM LDOS (see Fig. 12) there are no bonding or antibonding patterns for the two acceptor states, in agreement with the results of the toy model. Rather, a spatially symmetric separation seems to occur for some of the Mn pairs, in particular for larger distances along the 〈 100 〉 , 〈 211 〉 and 〈 111 〉 directions. For these pairs the upper and lower acceptor acquire opposite and rather definite spin characters in the easy direction. In this case spin up is located on one site and spin down on the other, whereas for 〈 110 〉 the two acceptors are of mixed 

<img src='content_image/426016.jpg'>

spin character. 

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

In this Section we take a closer look at the acceptor energy splitting Δ$_{acc}$( ˆ Ω) ≡ ε$_{2}$ − ε$_{1}$ , focusing in particular on its spherical average ˜ Δ$_{acc}$ = ˜ ε$_{2}$ − ˜ ε$_{1}$ . We will also compute the effective exchange constant J ( ˆ Ω) and see if a relationship can be found between these two important quantities. 

Fig. 13 shows the average acceptor splitting ˜ Δ$_{acc}$ in the FM and AFM configurations compared to the experimen- tal and theoretical values reported in Ref. [4,5]. Our cal- culated splittings match the previous calculations $^{4,5}$, and follow the trend of the measured splittings at the (110) surface. Fig. 13 (b) reveals that the splitting is largest for Mn pairs along 〈 110 〉 directions, and very small for 〈 100 〉 Mn pairs. The 〈 211 〉 and 〈 111 〉 Mn pairs exhibit smaller splittings than 〈 110 〉 , which has more direct hop- ping paths. Notice that there is some correlation between the splittings and the high barriers in Fig. 9 (a); higher 

