The possibility of having positronium (Ps) physisorbed at a material surface is of great fundamental interest, since it can lead to new insight regarding quantum sticking and is a necessary first step to try to obtain a Ps$_2$ molecule on a material host. Some experiments in the past have produced evidence for physisorbed Ps on a quartz surface, but firm theoretical support for such a conclusion was lacking. We present a first-principles density-functional calculation of the key parameters determining the interaction potential between Ps and an $alpha$-quartz surface. We show that there is indeed a bound state with an energy of 0.14 eV, a value which agrees very well with the experimental estimate of $sim0.15$ eV. Further, a brief energy analysis invoking the Langmuir-Hinshelwood mechanism for the reaction of physisorbed atoms shows that the formation and desorption of a Ps$_2$ molecule in that picture is consistent with the above results.
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