Orbital-Based Correlated Electron–Nuclear Dynamics for Extended Systems with Exact Factorization

In this work, we introduce a practical orbital-based framework for simulating correlated electron-nuclear dynamics in extended systems within the exact factorization (XF) formalism. Building on our earlier derivation of time-dependent Kohn-Sham (TDKS) equations that merge real-time time-dependent density functional theory with XF, we apply the classical path approximation and incorporate pairwise XF-derived decoherence corrections in the Kohn-Sham basis. This leads to a new efficient algorithm capable of treating nonadiabatic processes involving thousands of atoms. As a demonstration, we perform nonadiabatic dynamics simulations of two spiro-type hole-transport materials under periodic boundary conditions─the first application of XF-based methods to extended systems. While hole dynamics without decoherence yield unphysical, persistent coherences, the inclusion of XF-derived decoherence produces physically consistent relaxation from lower to higher bands.