Deep-Dive Guides · Cages & Metal Clusters

Metal Clusters & Cages: Electron-Counting, M–M Bonding, Boranes, Carboranes & Zintl Ions

A single-file deep dive into the five electron-counting frameworks that govern clusters and cages — the 18-electron/EAN rule, Wade–Mingos PSEPT, cluster-valence-electron counting, the isolobal analogy, and the Zintl–Klemm concept — with worked counts, fully-illustrated polyhedra, graded problems, and CSIR Part-C practi.

Free extract — the opening section of the guide, in full.

0 · How to use this guide & a map of the frameworks ▲ top

Cluster and cage chemistry is not one theory but a toolbox of electron-counting frameworks. The single most common exam error is applying the wrong framework to a given species. This guide is organised so that each framework is introduced, worked, and then contrasted with its neighbours in the "Look-alike traps" section. Read the map below first, keep it open while you work the problems, and always finish a count by asking "which framework did the question intend?"

The golden workflow. (1) Identify the species type — mononuclear complex, metal carbonyl cluster, main-group cage (borane/carborane), or naked p-block anion (Zintl). (2) Pick the matching framework from the table. (3) Count carefully, writing every contribution. (4) Translate the number into a structure (bond order, polyhedron, or closo/nido/arachno class). (5) Cross-check against a magic number or parent deltahedron.

The five frameworks and when each applies

Framework	What it counts	Best used for	Key output
18-electron / EAN rule	Total valence electrons at one metal (metal d-electrons + ligand donations)	Mononuclear organometallics; each metal in small clusters (with M–M bonds counted)	Stability / electron sufficiency at a metal
CVE (cluster valence electrons) / Mingos TEC	Total valence electrons of the whole metal cluster	Transition-metal carbonyl clusters (triangles, tetrahedra, octahedra…)	Number of M–M bonds; magic-number polyhedron
Wade's rules / PSEPT	Skeletal electron pairs (SEP) used in cage bonding only	Boranes, carboranes, and (via isolobal) main-group & metal clusters	closo / nido / arachno / hypho class → geometry
Isolobal analogy (Hoffmann)	Frontier-orbital equivalence between fragments	Swapping BH ↔ M(CO)₃, CH ↔ Co(CO)₃ etc. so Wade applies to mixed clusters	Predicting analogous structures
Zintl–Klemm concept	Electron transfer from electropositive metal to a p-block (pseudo-)element network	Naked polyanions [Sn₉]⁴⁻, Zintl phases (NaTl)	Network topology (8−N rule) or Wade class for closo anions

Note on terminology. "EAN" (effective atomic number) and the "18-electron rule" are the same idea expressed two ways. EAN counts every electron on the metal and asks for the next noble-gas number: 36 (Kr) for the first transition series, 54 (Xe) for the second, 86 (Rn) for the third. Because the closed core beneath the valence shell holds 18, 36 and 68 electrons respectively, reaching the noble-gas number is arithmetically identical to reaching 18 valence electrons at the metal (36−18 = 18, 54−36 = 18, 86−68 = 18). Either language is acceptable in an answer; this guide uses the 18-electron form because it extends cleanly to CVE counting.

Confirmed, with one scope limit worked out in §5. The Wade–Mingos–Lauher extension is standard, and the skeletal contribution of a transition-metal vertex is v + x − 12 (v = group number, x = electrons donated by the ligands on that metal). The “−12” is not universal, however: it assumes the metal devotes 3 of its 9 valence orbitals to cage bonding. For a fragment using f frontier orbitals the general form is (v + x) − 18 + 2f, which collapses to v + x − 12 only when f = 3.

Read the rest of this guide

The remaining 10 sections — Metal–metal bonding & the quadruple bond, Electron-counting I — 18-electron/EAN rule & the CVE approach, Wade's rules / PSEPT for boranes, Boranes & carboranes in depth and more — are part of ChemVidya Full Access, along with the other guides, the books, the study notes and 6,000+ practice questions with worked solutions.

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