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In our discussion of the toy model, we have seen that the emergence of a stable FM configuration for the Mn pair is brought about by the “widening” of a partially occupied cluster of impurity levels caused by hopping. It is therefore instructive to compare some measures of the “bandwidth” of the six impurity levels with J . Fig. 17 (a) shows the FM effective bandwidth of the six impurity levels, 

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and Fig. 17 (b) shows the splitting between the mean of the four occupied impurity levels and the mean of the two unoccupied levels, 

In the FM configuration these can both be seen as a mea- sure of the effective exchange interaction strength. A comparison of Figs. 17 and 14 reveals that they both do quite well in reproducing the correct trend in J , with the exception of the most closely spaced 〈 110 〉 pair. These observations support the validity of the simplified argu- ments used in the toy model. 

In conclusion, our analysis of the acceptor splitting versus effective exchange constant shows that these two quantities are certainly related but not in a direct quan- titative way. In particular, the acceptor splitting is much more anisotropic than J as a function of the Mn pair ori- entation. Other quantities such as w and δ [see Eqs. (9) and (10)] involving all six impurity levels seems have a much better correspondence with J . It is interesting to ask whether or not the four occupied impurity levels are also accessible by STM. One might naively expect that when the applied bias is reversed, electrons tunneling out of the occupied impurity levels should appear as sharp features in the differential conductance spectrum. The experiments of Ref. [4] do not show any clear sign of these levels. However, a more recent study 34 on individual Mn impurities in GaAs indicates that a weak feature due to tunneling out of one the three t$_{2}$$_{g}$ levels can indeed be seen below the top of the valence band. Presumably this is due to a hole-hole interaction effect: the final state has two electrons missing near the Mn impurity. This feature is very hard to see, as it is masked by electrons tunneling out of the valence band continuum. 

## 4. Comparison with ab initio estimates of the exchange coupling J 

In this section we compare our results for the effec- tive exchange J with those of previous studies. Since the pair exchange interaction between impurities is a cru- cial quantity in the field of DMS, there have been many theoretical studies of this quantity, mainly based on first- principles calculations. Here we concentrate our atten- tion on a few issues that have emerged from our theoret- ical approach: (i) the relatively large value for the 〈 110 〉 pair with shortest Mn spacing; (ii) the non-monotonic de- cay of J with Mn separation and its strongly anisotropic character with respect to pair orientation; (iii) the com- parison with acceptor splitting, with particular reference to the discrepancies between J and Δ$_{acc}$ found in our model, e.g., for the 〈 100 〉 pairs. A few caveats are nec- essary before comparing our results with other estimates of J that have appeared in the literature. Firstly, most of the published estimates are for much higher Mn con- centrations, which strongly affects the value of J and its 

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