  We consider the interaction between acceptor pairs in doped semiconductors in
the limit of large inter-acceptor separation relevant for low doping densities.
Modeling individual acceptors via the spherical model of Baldereschi and
Lipari, we calculate matrix elements of the quadrupole tensor between the four
degenerate ground states and show that the acceptor has a nonzero quadrupole
moment. As a result, the dominant contribution to the large-separation
acceptor-acceptor interaction comes from direct (charge-density) terms rather
than exchange terms. The quadrupole is the leading nonzero moment, so the
electric quadrupole-quadrupole interaction dominates for large separation. We
calculate the matrix elements of the quadrupole-quadrupole interaction
Hamiltonian in a product-state basis and diagonalize, obtaining a closed-form
expression for the energies and degeneracies of the sixteen-state energy
spectrum. All dependence on material parameters enters via an overall
prefactor, resulting in surprisingly simple and universal results. This
simplicity is due, in part, to a mathematical happenstance, the nontrivial
vanishing of a particular Wigner 6-j symbol. Results are relevant to the
control of two-qubit interactions in quantum computing implementations based on
acceptor spins, as well as calculations of the thermodynamic properties of
insulating p-type semiconductors.
