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The energetics and atomic structure associated with the localized hole formed near an Al-atom dopant in ?-quartz are calculated using a variational, self-consistent implementation of the Perdew?Zunger self-interaction correction with complex optimal orbitals. This system has become an important test problem for theoretical methodology since generalized gradient approximation energy functionals, as well as commonly used hybrid functionals, fail to produce a sufficiently localized hole due to the self-interaction error inherent in practical implementations of Kohn?Sham density functional theory. The self-interaction corrected calculations are found to give accurate results for the energy of the defect state with respect to both valence and conduction band edges as well as the experimentally determined atomic structure where only a single Al?O bond is lengthened by 11%. The HSE hybrid functional, as well as the PW91 generalized gradient approximation functional, however, gives too small an energy gap between the defect state and the valence band edge, overly delocalized spin density and lengthening of more than one Al?O bond.

Calculations of Al dopant inα-quartz using a variational implementation of the Perdew–Zunger self-interaction correction