Alkali metal gold bismuthides, A2AuBi, are isoelectronic with alkali metal thallides, ATl = A2TlTl, and yet Na2AuBi adopts an orthorhombic structure with a 1-D zigzag “ribbon” structural motif rather than the cubic double diamond structure type of NaTl as well as Li2AuBi. Using first principles quantum mechanical calculations applied to A2AuBi, hypothetical “A2HgPb,” and A2TlTl, and comprehensively decomposing the total energies into metallicity, ionicity, and covalency components to establish parallels with the qualitative Zintl–Klemm formalism, the factors determining the relative stability between the zigzag “ribbon” and the diamond network are examined. An interplay between volume-dependent energy terms, i.e., metallicity or ionicity, and covalency among the electronegative components determines which structural motif is favored. In Na2AuBi, there are two factors stabilizing the zigzag “ribbon.” Au 5d states significantly interact with Bi 6p states, especially Au 5d with Bi 6pz to promote stronger Au–Bi covalent interactions than in the diamond network. This factor does not exist in Na2TlTl and “A2HgPb,” where Hg, Tl, and Pb 5d states are well localized. Secondly, the zigzag ribbons provide effective covalent interactions at larger volumes, as in Na2AuBi, while effective covalent interactions occur in the diamond network only at smaller volume, as in Li2AuBi.