Shadow excited state molecular dynamics with the ΔSCF method

We present an extension of the shadow extended Lagrangian Born-Oppenheimer molecular dynamics method to excited state molecular dynamics (ESMD) in the context of ΔSCF Kohn-Sham density functional theory, with demonstrations performed using self-consistent charge density functional tight binding (SCC-DFTB) theory. In this shadow ESMD approach, the approximate iterative solution to the exact potential in conventional ESMD is replaced by an exact single-step solution to an approximate shadow excited-state potential. The energy functional that defines this shadow excited-state potential as a stationary (non-aufbau) solution is obtained from a linearization about an approximate excited state density, which would become a progressively worse approximation as the dynamics ensue if it were static. To avoid this, we propagate the approximate excited-state (charge) density as an additional dynamical variable in an extended Lagrangian approach. We show that, in addition to offering significant improvement in computational cost relative to direct ESMD, our shadow ESMD method provides enhanced stability and robustness relative to its "exact" counterpart. Our implementation is carried out in the context of SCC-DFTB theory but should be broadly generalizable, both to ab initio electronic structure methods and to other semi-empirical quantum chemistry approaches.