Extended Hellmann–Feynman forces, canonical representations, and exponential propagators in the mixed quantum-classical molecular dynamics

Abstract

Analytical expressions of the Hellmann–Feynman (HF) forces in the quantum-classical molecular dynamics (QCMD) are evaluated and analyzed. The conventional expression of the HF forces is valid in the differential form of the QCMD evolution equations, but the extended formula appears in the context of approximate, time-step propagators. The canonical Hamilton representation of QCMD, and its symplectic and nonsymplectic exponential propagators, are reviewed. Tests for a model proton transfer system are performed in order to compare efficiency of the proposed integration schemes. The most efficient scheme results from separation of either different time scales or different approximation orders for the quantum and classical parts, and also from correct accumulation of the HF forces, corresponding to an improved extended HF formula. We derive the canonical representation and propagators of QCMD in the adiabatic basis set. If the classical and quantum parts of the propagator are separated in that representation, the extended HF forces appear, and are related to transitions between the adiabatic states. Applications to the quantum-classical molecular dynamics are proposed, using multiple protonic and/or electronic adiabatic states.