Non-Born-Oppenheimer Treatment of Intramolecular Proton Transfer: Effects of Gradient-Dependent Electron-Proton Correlation on Tunnel Splitting Energy in Malonaldehyde

Standard treatments of proton tunneling rely on potential energy surfaces generated under the Born–Oppenheimer approximation, with tunneling introduced as a subsequent correction. We apply Nuclear-Electronic Orbital Multistate Density Functional Theory (NEO-MSDFT) to malonaldehyde, treating the transferring proton as a quantum mechanical degree of freedom that evolves simultaneously with the electronic density. We examine how the gradient-dependent electron-proton correlation functional (epc19) modifies the calculated tunnel splitting energy. The results indicate that explicit electron-proton correlation is required to maintain physically consistent proton delocalization and to recover a finite tunnel splitting. Malonaldehyde serves as a well-characterized benchmark system whose experimental ground-state splitting (∼21.6 cm⁻¹) and deuterium-substituted splitting (∼2.9 cm⁻¹) provide clear validation targets. The molecular framework is interpreted as a localized dissipative environment whose vibrational modes couple to the thermal bath, providing a mechanism for entropy removal that protects the coherence of the protonic superposition. The primary theoretical advance is the framing of the molecular skeleton as a Phonon Sink that irreversibly removes entropy from the tunneling coordinate via Landauer’s principle, thereby identifying a built-in coherence-protection mechanism relevant to protected nuclear qudits and engineered dissipative quantum systems.