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$\alpha$-graphyne is a two-dimensional sheet of $sp$-$sp^2$ hybridized carbonatoms in a honeycomb lattice. While the geometrical structure is similar tothat of graphene, the hybridized triple bonds give rise to electronic structurethat is different from that of graphene. Similar to graphene, $\alpha$-graphynecan be stacked in bilayers with two stable configurations, but the differentstackings have very different electronic structures: one is predicted to havegapless parabolic bands and the other a tunable band gap which is attractivefor applications. In order to realize applications, it is crucial to understandwhich stacking is more stable. This is difficult to model, as the stability isa result of weak interlayer van der Waals interactions which are not wellcaptured by density functional theory (DFT). We have used quantum Monte Carlosimulations that accurately include van der Waals interactions to calculate theinterlayer binding energy of bilayer graphyne and to determine its most stablestacking mode. Our results show that interlayer bindings of $sp$- and$sp^{2}$-bonded carbon networks are significantly underestimated in a Kohn-ShamDFT approach, even with an exchange-correlation potential corrected to include,in some approximation, van der Waals interactions. Finally, our quantum MonteCarlo calculations reveal that the interlayer binding energy difference betweenthe two stacking modes is only 0.9(4) meV/atom. From this we conclude that thetwo stable stacking modes of bilayer $\alpha$-graphyne are almost degeneratewith each other, and both will occur with about the same probability at roomtemperature unless there is a synthesis path that prefers one stacking over theother.

The Nature of Interlayer Binding and Stacking of $sp$-$sp^{2}$ Hybridized Carbon Layers: A Quantum Monte Carlo Study