Real-Space and Real-Time Propagation for Correlated Electron-Nuclear Dynamics Based on Exact Factorization

We present coupled equations of motion for correlated electron-nuclear dynamics for real-space and real-time propagation with a proper electron-nuclear correlation (ENC) from the exact factorization. Since the original ENC term from the exact factorization is non-Hermitian, the numerical instability arises as we propagate an electronic wave function. In this paper, we propose a Hermitian-type ENC term which depends on the electron density matrix and the nuclear quantum momentum. Moreover, we show that the Hermitian property of the electron- nuclear correlation term can capture quantum (de)coherence with a stable numerical real-space and real-time propagation. As an application, we demonstrate a real-space and real-time propagation of an electronic wave function coupled to trajectory-based nuclear motion for a one-dimensional model Hamiltonian. Our approach can capture nonadiabatic phenomena as well as quantum decoherence in excited state molecular dynamics. In addition, we propose a scheme to extend the current approach to many-body electronic states based on real-time time-dependent density functional theory, testing the nonadiabatic dynamics of a simple molecular system.