High-order geometric integrators for representation-free Ehrenfest dynamics

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 the norm of 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.

NQCDynamics.jl: A Julia package for nonadiabatic quantum classical molecular dynamics in the condensed phase

“…• Classical molecular dynamics 66 • Molecular dynamics with electronic friction (MDEF) 43,44 • Ehrenfest molecular dynamics [7][8][9] • Fewest-switches surface hopping 49 • Ring polymer molecular dynamics (RPMD) 75,76 • Nonadiabatic RPMD (NRPMD) [35][36][37][38] • Centroid ring polymer surface hopping (RPSH) 14,15 • Centroid ring polymer Ehrenfest dynamics 77 • Extended classical mapping model (eCMM) 30,78 • Generalized spin mapping approach 79,80 C. Defining the Hamiltonian with NQCModels.jl…”

“…As an alternative to surface hopping dynamics, a mean-field approach can be taken such that the force is averaged over all states, weighted by the electronic populations. [7][8][9] The Ehrenfest Hamiltonian can be written as…”

“…A variety of mixed quantum-classical and semiclassical dynamics methods have been developed over the years that retain an (approximate) description of quantum effects while providing improved computational scaling properties. Examples include: Ehrenfest dynamics, [7][8][9] molecular dynamics with surface hopping, [10][11][12][13][14][15][16] mixed quantum-classical Liouville dynamics, [17][18][19][20] the quantum-classical path inte-gral method, 21,22 and semiclassical mapping Hamiltonian methods. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] Most of these methods were conceived with a relatively small number of electronic states in mind, but some have been extended and modified to tackle the continuum of states encountered in metallic environments.…”

NQCDynamics.jl: A Julia Package for Nonadiabatic Quantum Classical Molecular Dynamics in the Condensed Phase

“…• Classical molecular dynamics 67 • Molecular dynamics with electronic friction (MDEF) 43,44 • Ehrenfest molecular dynamics [7][8][9] • Fewest-switches surface hopping 49 • Ring polymer molecular dynamics (RPMD) 74,75 • Nonadiabatic RPMD (NRPMD) [35][36][37][38] • Centroid ring polymer surface hopping (RPSH) 14,15 • Centroid ring polymer Ehrenfest dynamics 76 • Extended classical mapping model (eCMM) 30,77 • Generalized spin mapping approach 78,79 C. Defining the Hamiltonian with NQCModels.jl…”

“…A variety of mixed quantum-classical and semiclassical dynamics methods have been developed over the years that retain an (approximate) description of quantum effects while providing improved computational scaling properties. Examples include: Ehrenfest dynamics, [7][8][9] molecular dynamics with surface hopping, [10][11][12][13][14][15][16] mixed quantum-classical Liouville dynamics, [17][18][19][20] the quantum-classical path integral method, 21,22 and semiclassical mapping Hamiltonian methods. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] Most of these methods were conceived with a relatively small number of electronic states in mind, but some have been extended and modified to tackle the continuum of states encountered in metallic environments.…”