The classical theory of chemical valence, first, is naturally formalized in mathematics in the area of graph theory and, second, finds an extension in quantum mechanics in terms of the Heitler-London-Pauling "valence-bond" (VB) theory. Thus, VB theory stands in a fairly unique position, although in quantum chemistry, there often has been a preference for the alternative (perhaps even "complementary") molecular orbital (MO) theory, presumably in large part because of computational efficacy for general molecular structures. Indeed, as formulated by Pauling and others, VB theory was described as a configuration interaction (CI) problem when there were multiple relevant classical valence structures for the same molecular structure. Also, as now recognized, a direct assault on CI is computationally intensive, prone to size-inconsistency problems, and effectively limited to smaller systems-whereas indirect approaches, e.g., via wave-function cluster expansions or renormalization-group theory, often neatly avoid these problems. Thus, what is (and perhaps always has been) needed is "many-body" schemes for VB computations (as well as for higher-order MO-based approaches, too). Here, then, certain such many-body VB-amenable computational schemes are to be discussed, in the context of semiempirical (explicitly correlated) graphical models. The collection of models are described and interrelated in a fairly comprehensive systematic manner. A selection of many-body cluster expansion methods are then discussed with special reference to resonating VB wave functions and the fundamental graph-theoretic nature of the consequent problems (such as also are noted to arise in lattice-discretized statistical-mechanical problems, too). Some examples are described incorporating resonance among exponentially great numbers of VB structures as applied: for large icosahedral-symmetry fullerenic structures, for the (polyacetylenic) linear chain, and for ladderlike conjugated polymers. It is contended that practicable many-body VB-theoretic methods are now available, retaining clear links to classical chemical valence theory. Hopefully, too, these methods may soon find use beyond the semi-empirical framework. 1997 John Wiley & Sons, Inc.
