Nuclear Fukui functions and Marcus reorganization: from vertical force response to diabatic work

Marcus theory expresses electron-transfer activation in terms of diabatic nuclear reorganization, whereas conceptual density functional theory (CDFT) characterizes electronic-nuclear response through electron-number derivatives. We connect these descriptions through the nuclear Fukui function (NFF), Φ = (∂F/∂N)_v. For exact ensemble DFT between adjacent integer charge states, the one-sided NFF at fixed geometry equals the finite difference of the corresponding nuclear-force fields. Treating that quantity as a force field over nuclear configuration space yields , which is independent of harmonicity and, because the field is conservative, of the integration path. In the symmetric Marcus limit this becomes . The familiar compliance expression, e.g., follows only after a local harmonic approximation. Gas-phase calculations on N₂/N₂⁺, NH₃/NH₃⁺, naphthalene/naphthalene^-, and cyclooctatetraene (COT)/COT^- illustrate this hierarchy. The quadratic reduction gives small absolute errors for N₂, although the state-specific relative deviations are about 6%; for naphthalene it gives a good total reorganization sum through compensating state-specific deviations of about 10%, deteriorates for the large-amplitude NH₃ umbrella relaxation, and fails strongly for COT, where a 99.35 cm^-1 out-of-plane puckering/torsional mode accounts for 96.32% of the forward local-compliance estimate. In contrast, the finite force-work integral remains accurate even when the local quadratic reduction fails: with 21 sampled geometries it reproduces λ_R + λ_P to 2.2 × 10^-4% for naphthalene and to 0.05% for COT. We distinguish the exact adjacent-charge-state construction from donor-acceptor electron transfer at fixed total electron number, for which fragment-population or explicitly diabatic electronic coordinates are required.