Analytical Gradients for Nuclear–Electronic Orbital Time-Dependent Density Functional Theory: Excited-State Geometry Optimizations and Adiabatic Excitation Energies

The computational investigation of photochemical processes often entails the calculation excited-state geometries, energies, and energy gradients. nuclear–electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on same level as electrons, thereby including associated nuclear effects non-Born–Oppenheimer behavior into chemistry calculations. multicomponent density functional theory (NEO-DFT) time-dependent DFT (NEO-TDDFT) methods allow efficient calculations ground excited states, respectively. Herein, analytical gradients are derived implemented for NEO-TDDFT method Tamm–Dancoff approximation (NEO-TDA). programmable equations these well NEO-DFT Hessian provided. NEO includes anharmonic zero-point (ZPE) delocalization with protons vibronic mixing in geometry optimizations states. harmonic ZPE other nuclei can be computed via Hessian. This is used to compute 0–0 adiabatic excitation energies a set nine small molecules all quantized, exhibiting slight improvement over conventional electronic approach. Geometry two intramolecular proton-transfer systems, [2,2′-bipyridyl]-3-ol [2,2′-bipyridyl]-3,3′-diol, performed one quantized systems produce electronically geometries stronger hydrogen bonds similar relative stabilities compared methods. work provides foundation nonadiabatic dynamics simulations fundamental such photoinduced proton transfer proton-coupled electron transfer.