Real-Time Propagation via Time-Dependent Density Functional Theory Plus the Hubbard U Potential for Electron–Atom Coupled Dynamics Involving Charge Transfer

We present methods for combining time-dependent density functional theory and the Hubbard U potential in the framework of the real-time propagation of Kohn–Sham orbitals to describe electron–atom coupled dynamics beyond the Born–Oppenheimer approximation. The time evolution of the noncommuting nonlocal operators were realized through Crank–Nicolson’s inversion method and Suzuki–Trotter’s split exponentiation. The electron dynamics related to the high speed motion of an alkali atom on a conjugated carbon plane is presented. The nonequilibrium charge oscillation between a metal surface and a localized atomic orbital, as modeled with graphene and Ca, is discussed.