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trace of the p - d exchange operator in Eq. (1) summed over all valence band states including the acceptors is zero. In the FM variation of the Mn spins, the high- energy acceptor level, ε$_{2}$ ( ˆ Ω), varies very little and in a qualitatively opposite manner to the low-energy accep- tor, ε$_{1}$ ( ˆ Ω). Its effect is therefore to reduce the much higher anisotropy coming from the lower acceptor level. In the AFM configuration small acceptor level variations partially cancel to a low total anisotropy below 1.5 lattice constants, and add up to a larger total anisotropy above this value. 

SO interaction mixes spin components, such that the eigenstates are no longer of definite spin character. As a result, the acceptor states above the valence band edge, which without SO coupling have the same spin charac- ter as the corresponding localized Mn moment (”spin- up”), acquire a small component in the opposite direc- tion (“spin-down”). In the FM cases, this results in a small spin down character of the acceptor levels, that in- creases with Mn distance primarily for the low acceptor, as it moves closer to the valence band. In the AFM varia- tions, the two acceptor levels are now very close in energy and their spin character can vary by a large amount on the unit sphere due to spin-orbit interaction. 

In Fig. 10 we plot the energies of the four highest oc- cupied and the two unoccupied acceptor levels for all studied configurations. We consider spherical averages of these six energies over all possible moment directions, denoted by ˜ ε$_{i}$ , i = − 3 , . . . , 2. 

In the FM configuration [Fig. 10 (a)] we find that there is a clear splitting between the three lowest and highest levels, in agreement with the results of the toy model shown in Fig. 5. The fact that three highest levels are also spread over a sizable energy range indicates that, at least for the shortest Mn separation, we are in the regime described by Fig. 6 of the toy model. Note in particular the large splitting of the two acceptor levels, as expected for aligned Mn spins. The level splitting and the overall width of the impurity band 50 decreases with increasing Mn separation. Note also the strong dependence of the acceptor splitting on the orientation of the Mn pair. In particular, the FM acceptor splittings tend to be largest along the 〈 110 〉 direction. It is also clear that the 〈 100 〉 Mn pair behaves differently. Even if the splitting is large between the three lower and three upper levels, the split- ting of the two acceptor levels is consistently smaller than for the other orientations. 

In the AFM configuration the structure of the six en- ergy levels is quite different and its salient features are nicely captured by the toy model result of Fig. 3. The levels are always essentially doubly degenerate; the split- ting between doublets can be large for the shortest Mn separation and but it decreases quickly with distance and becomes of the order of the expected SO-induced split- ting. In contrast to the FM case, there is no visible split- ting between the acceptor levels. The only exception is the special case of the 〈 110 〉 for the shortest Mn separa- tion, where the six impurity levels abruptly drop down 

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towards the valence continuum as the off-site Coulomb correction on the common As neighbor is decreased. Fi- nally note that the 〈 100 〉 pair sticks out with a more dense set of levels also in the AFM configuration 

We proceed to discuss the Local Density Of States (LDOS) of the two acceptor levels - a property that can be probed by STM spectroscopy. 4–7 Plots of the LDOS of the two acceptor levels in the (110) plane containing the two Mn, are shown in Fig. 11 for the FM and in Fig. 12 for the AFM configurations. The LDOS plots are generated 54 from the tight-binding model calculations as in Refs. [10,33]. In these plots the Mn spins are pointing in the magnetic easy directions, indicated by the arrow in the top left corner of each panel. The left column in each figure shows the LDOS of the lower energy acceptor state, which is closer to the valence band maximum; the right column refers to the acceptor with higher energy. 

