We construct a theory for electron-rich polyanionic networks in the intermetallic compounds of heavy late main group elements, building a bonding framework that makes a connection to well-understood hypervalent bonding in small molecules such as XeF(4), XeF(2), and I(3)(-). What we do is similar in spirit to the analogy between the Zintl-Klemm treatment of classical polyanionic networks and the octet rule for molecules. We show that the optimal electron count for a linear chain of a heavy main group element is seven electrons per atom, six electrons per atom for a square lattice, and five electrons per atom for a simple cubic lattice. Suggestions that these electron counts are appropriate already exist in the literature. We also derive electron counts for more complicated topologies, including one-dimensional ladders and one dimensional strips cut from a square lattice. We also study pairing (Peierls) distortions from these ideal geometries as well as other deformations. The presence of s-p mixing (or its absence) plays a critical role in the propensity for pairing and, in general, in determining the geometrical and electronic structure of these phases. Hypervalent bonding goes along with the relative absence of significant s-p interaction; there is a continuum of such mixing, but also a significant difference between the second-row and heavier elements. We attribute the existence of undistorted metallic networks of the latter elements to diminished s-p mixing, which in turn is due to the contraction of less-screened s orbitals relative to p orbitals down the groups in the Periodic Table. The number of electrons in the polyanionic network may be varied experimentally. An important general principle emerges from our theoretical analysis: upon oxidation a hypervalent structure transforms into a classical one with the same lattice dimensionality, while upon Peierls distortion the hypervalent structures transform into classical ones with the lattice dimensionality reduced. Dozens of crystal structure types, seemingly unrelated to each other, may be understood using the unifying concept of electron-rich multicenter bonding. Antimonides, which are explored in great detail in the current work, conform particularly well to the set of electron counting rules for electron-rich nonclassical networks. Some deviation up and down from the ideal electron count is exhibited by known stannides and tellurides. We can also make sense of the bonding in substantially more complicated alloys, including La(12)Mn(2)Sb(30) and Tl(4)SnTe(3). The hypervalent electron counting scheme developed in this paper, along with the classical Zintl-Klemm electron counting rules, gives an easy qualitative understanding of bonding in a wide variety of intermetallic compounds of heavy main group elements.