Abstract
Ehrenfest dynamics is a useful approximation for ab initio mixed quantum-classical molecular dynamics that can treat electronically nonadiabatic effects. Although a severe approximation to the exact solution of the molecular time-dependent Schrödinger equation, Ehrenfest dynamics is symplectic, is time-reversible, and conserves exactly the total molecular energy as well as thenormof the electronic wavefunction. Here, we surpass apparent complications due to the coupling of classical nuclear and quantum electronic motions and present efficient geometric integrators for "representation-free" Ehrenfest dynamics, which do not rely on a diabatic or adiabatic representation of electronic states and are of arbitrary even orders of accuracy in the time step. These numerical integrators, obtained by symmetrically composing the second-order splitting method and exactly solving the kinetic and potential propagation steps, are norm-conserving, symplectic, and time-reversible regardless of the time step used. Using a nonadiabatic simulation in the region of a conical intersection as an example, we demonstrate that these integrators preserve the geometric properties exactly and, if highly accurate solutions are desired, can be even more efficient than the most popular non-geometric integrators.
Highlights
Mixed quantum-classical methods, such as surface hopping,1–5 mean-field Ehrenfest dynamics,6–15 and methods based on the mixed quantum-classical Liouville equation16–18 or the Meyer–Miller–Stock–Thoss mapping Hamiltonian,19–25 remedy one of the shortcomings of classical molecular dynamics: its inability to describe electronically nonadiabatic processes26–28 involving significantly coupled29–31 states
Almost every geometric property55–57 can, be preserved exactly by employing the scitation.org/journal/jcp symplectic integrators58 based on the splitting method.59,60. This splitting method is widely applicable—so long as the Hamiltonian can be decomposed into exactly solvable parts—and was employed to obtain symplectic integrators in many wellknown applications, including molecular quantum61 and classical62 dynamics, Schrödinger–Liouville–Ehrenfest dynamics,63 and the Meyer–Miller–Stock–Thoss mapping approach
The time-dependent Hartree (TDH)57,80–84 approximation is an optimal approximate solution to the molecular timedependent Schrödinger equation (TDSE) (1) among those in which the molecular state can be written as the Hartree product
Summary
Mixed quantum-classical methods, such as surface hopping, mean-field Ehrenfest dynamics, and methods based on the mixed quantum-classical Liouville equation or the Meyer–Miller–Stock–Thoss mapping Hamiltonian, remedy one of the shortcomings of classical molecular dynamics: its inability to describe electronically nonadiabatic processes involving significantly coupled states. The widely used two and three time step methods improve the efficiency by using different integration time steps that account for the different time scales of nuclear and electronic motions (see Appendix A) Such integration schemes violate the geometric properties of the exact solution: the simpler, two time step method is irreversible and neither method is symplectic (see Fig. 6 in Appendix A). Scitation.org/journal/jcp symplectic integrators based on the splitting method.59,60 This splitting method is widely applicable—so long as the Hamiltonian can be decomposed into exactly solvable parts—and was employed to obtain symplectic integrators in many wellknown applications, including molecular quantum and classical dynamics, Schrödinger–Liouville–Ehrenfest dynamics, and the Meyer–Miller–Stock–Thoss mapping approach..
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