This paper analyzes the importance of s,p mixing-a necessary addition to the simplest Rundle-Pimentel picture-and periodic and group trends in electron-rich three-center bonding. Our analysis proceeds through a detailed quantum chemical study of the stability of electron-rich three-center bonding in triatomic 22-valence electron anions, To provide interpretations, a perturbational molecular orbital (MO) analysis of s,p mixing is carried out. This analysis of the orbitals and the overlap populations is then tested by density functional calculations for a number of linear trihalides, trichalcogenides, and tripnictides. The most important effect of s,p mixing on the in-line bonding is in destabilization of the 3sigma(g) orbital and is determined by the overlap between the s orbital of the central atom and the p orbital of the terminal atom. Further destabilization arises from the repulsion of p, lone pairs. Both of these antibonding effects increase with increasing negative charge of the system. The stability of isoelectronic X-3 systems thus decreases when moving from right to left in the periodic table. Interesting group trends are discerned; for instance, for the electron-rich tripnictides, the ability to accommodate a hypervalent electron count is the largest in the middle rather at the end of the roup. Particularly strong s,p mixing can reverse the bonding/antibonding character of MOs: thus MO 2sigma(u) that is responsible for bonding for trihalides and trichalcogenides is actually antibonding in N-3(7-).