Does the zinc vacancy in ZnGeP$_2$ exhibit a Jahn-Teller distortion?

The Zn-vacancy is one of the dominant defects in ZnGeP$_2$. Its single negative charge state is EPR active. The hyperfine splitting shows that the unpaired electron is primarily localized on a pair of P atoms. In contrast, first-principles 64 atom supercell calculations using both the FP-LMTO and the VASP method of the $V_{Zn}^{-}$ state show that the defect maintains $S_4$ symmetry with the wave function spread equally over 4 P atoms. Here a group-theoretical analysis is presented. When including only the nearest neighbors, the system has $D_{2d}$ symmetry. While the one electron state of the unpaired electron is non-degenerate, a doubly degenerate $e$-state lies only about 10 meV below it. We show that a P-pairing distortion mode splits this $e$-state in two states which are even with respect to one of the mirrorplanes and odd with respect to the other and thus can only contain two of the P-dangling bonds. Calculations in which a pairing of P atoms is enforced while relaxing the remaining atoms confirm this model. Remaining puzzling aspects of this defect will be discussed.