12 anisotropy barriers correspond to a larger splitting. Fig. 13 (a) shows that our tight-binding model as well as a previous theory 4 based on a similar approach, sys- tematically underestimates the acceptor splitting mea- sured experimentally. This discrepancy could be due to inherent limited accuracy of the tight-binding method. However, it is also quite possible that part of the splitting seen experimentally is of Coulombic origin arising when electrons tunnel into the two different acceptor levels. In Fig. 13 (c) we see that the AFM splittings are very small compared to the FM ones, and are typically of the order of a few meV. As already noted above, the only exception is the 〈 110 〉 ∗ pair. Reducing the Coulomb parameter on the common As neighbor dramatically lowers the accep- tor energies [see Fig. 10 (b)], produces more extended wave functions (see Fig. 12), and increases mixing with valence band states. We can now consider the effective exchange energy de- fined as J ( ˆ Ω) = [ E AFM tot ( ˆ Ω) − E FM tot ( ˆ Ω)] / [2 · (5 / 2) $^{2}$], where the factors of 5 / 2 from the Mn spin magnitudes is ab- sorbed in J . A positive (negative) J implies that the Mn-Mn interactions are FM (AFM). As a result of the spin-orbit interaction, J ( ˆ Ω) is an anisotropic quantity, that is, it depends on ˆ Ω. The effective ˜ J as obtained by taking the spherical av- erages is shown in Fig. 14, where it can be seen that the FM configuration is always the most stable one, ˜ J > 0. Only for separations greater than two lattice constants, do the SO-induced fluctuations in J become comparable to the average value. Even then, ferromagnetism is al- ways favored. The maximal value of J generally occurs when the Mn spins are pointing along the pair axis. The comparison between the acceptor splitting ˜ Δ FM acc for the FM state [Fig. 13 (b)] and the exchange constant ˜ J [Fig. 14 (a)] shows that both quantities decay rapidly with Mn separation. However, there are noticeable differ- ences between them, out of which two are most obvious. First of all, the relative value of ˜ J for the 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ pair compared to all the other values is much larger than the corresponding value of ˜ Δ FM $_{acc}$. Secondly, ˜ Δ FM acc displays a much less monotonic decrease with Mn separation than ˜ J . In particular, the large dip for the 〈 100 〉$_{d}$$_{=}$$_{a,}$$_{2}$$_{a}$ pairs is totally absent in the plot for ˜ J . A similar dip for the 〈 100 〉$_{d}$$_{=2}$$_{a}$ pair is hinted but much less pronounced in the ˜ J plot. At this point it is useful to consider other quanti- ties that can shed light on the relationship between the effective exchange interaction and the acceptor lev- els. We first consider the effective exchange constant for a Mn pair when only one hole is present or, equiv- alently, when the lower acceptor is occupied by an elec- tron. The toy model results in Figs. 3 and 4 can help intuition and justify the rational of this choice: when an extra electron is added to these electronic configu- rations, only the topmost acceptor level is unoccupied in the FM state. In this case the acceptor splitting should be directly related to the energy gain stabilizing a FM state over the AFM state. The effect of occupy- ing the lowest acceptor is shown in Fig. 15, where we plot ˜ J$_{1hole}$ ≡ [ ˜ E AFM tot , 1hole − ˜ E FM tot , $_{1hole}$] / [2 · (5 / 2) $^{2}$], in which ˜ E AFM , FM tot , 1hole is the spherical average of the total ground state energy for a system with one Mn pair and only one hole, namely with one extra electron added. We can indeed see that now the dependence of ˜ J$_{1hole}$ on the crys- tal orientation and spacing of the Mn pair is qualitatively much more similar to the Δ FM acc of Fig. 13 (b), displaying the same large dips for the 〈 100 〉 pairs. In Fig. 16 we plot the spherical averages of the effective J as estimated by the FM/AFM difference of the sums of the 4 occupied impurity levels and the 2 unoccupied levels. Comparing with the “exact” ˜ J calculated by total energy difference of the full system of electrons (Fig. 14), we see that both approximations underestimate the value of ˜ J for the most closely spaced 〈 110 〉 Mn pair. For the rest of the points, using the four highest occupied levels overestimates the value of ˜ J approximately by a factor of two. Surprisingly, taking the difference of the sums of the two acceptors in the FM and the AFM configurations, gives a very good estimate of the “exact” ˜ J for the rest of the points in the plot of Fig, 14 (a). In order to compute the effective ˜ J for the two acceptors, one must invert the sign, because 2 unoccupied levels are used. FIG. 15: Effective exchange couplings for a system with two Mn and one hole. Adding one electron to the system, gives an effective exchange that is related to the acceptor splittings. FIG. 14: Effective exchange couplings. A ferromagnetic cou- pling is always favored as all spherical averages ˜ J > 0. 1.0 1.5 2.0 2.5 3.0 3.5 0 10 110 110 100 100 100 211 211 211 111 50 110 110 48 8 6 4 2 12 meV R a 1.0 1.5 2.0 2.5 3.0 3.5 0 10 110 110 100 100 100 211 211 211 111 110 110 54 8 6 4 2 12 14 meV R a