The virial theorem for potential curves V(R) is recast into a linear second-order differential equation in the inverse picture R(V), which provides a framework for correlating the inner and outer walls of the potential well. Leading solutions extended for scaling define a new class of five-parameter potentials in terms of Gauss hypergeometric functions. Curves are classified with two labels (p,q) related to Dunham spectroscopic constants and long-range threshold behavior. Special limiting cases include the Morse potential, the harmonic oscillator, and the 2n:n inverse-power generalization of the Lennard-Jones 12:6 and Kratzer-Coulomb 2:1 potentials. Empirical maps of (p,q) values from experimental molecular data reveal distinct clustering of points correlated to covalent, van der Waals and ionic bonding. Semiclassical quantization gives hypergeometric formulas for energy levels and RKR potentials. Threshold behavior of exact molecular curves is consistent with a linear combination of elementary inverse potentials, with the extended hypergeometric basis as a first approximation.