Born-Oppenheimer, Born-Huang, and exact factorization: quantum geometry and error in analytically transparent benchmark models

The phrase "potential energy surface" refers to several distinct objects. The Born-Oppenheimer construction gives a clamped-nucleus electronic eigenvalue, the single-surface Born-Huang construction adds the diagonal correction, and exact factorization gives a state-dependent exact scalar potential. These constructions answer different questions and should not be regarded as competing definitions of one universal surface. We compare them in two analytically transparent benchmark models. For Fern'andez's bilinearly coupled oscillators, the exact molecular spectrum, the Born-Oppenheimer and Born-Huang spectra, and the ground-state exact-factorization surface are obtained in closed form. We prove the ground-state ordering of these energies for every admissible mass ratio and coupling and show why it does not extend uniformly to excited states. The diagonal Born-Huang correction is identified with the mass-weighted quantum metric, and Fern'andez's sixth-order result is recast as a leading geometric error budget involving the metric and a gap-weighted spectral moment of the same derivative couplings. In a linear vibronic-coupling model the metric localizes at an avoided crossing while the total Fubini-Study length remains pi/2, separating the localization of electronic-state change from its total magnitude. Exact factorization is smooth for the nodeless ground state but becomes increasingly ill-conditioned, while remaining finite, when an excited-state nuclear marginal becomes small. These models separate approximation error, geometric correction, and conditioning in a form that can be checked directly.