Deriving the exact nonadiabatic quantum propagator in the mapping variable representation

Abstract

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting expression is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact propagator lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact propagator, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation

“…6,7 Over the past two decades, several methods for the simulation of nonadiabatic processes have been developed including exact quantum time-propagation, [8][9][10] the symmetrical quasi-classical windowing method, 11 mixed quantum-classical Liouville methods, [12][13][14] and surface hopping. [15][16][17][18][19][20][21][22] In addition, approximate path-integral based methods such as ring polymer molecular dynamics [23][24][25][26][27] and centroid molecular dynamics 28 have also been extended to nonadiabatic systems.…”

Thermal quantum time-correlation functions from classical-like dynamics

“…where we have noted that the adjoint of the Liouvillian is its negative [104], and that L Moy [e −βĤ ] W (p, q) = 0 since exact quantum dynamics conserves the quantum Boltzmann distribution. We illustrate Eq.…”

Path integral molecular dynamics with surface hopping for thermal equilibrium sampling of nonadiabatic systems

“…In this test, we use the test potential (24), with β = 1, β N = 1 16 , M = 10 and η = 1, and vary the time step sizes for the numerical integration. We choose the same diagonal observable as before in (25). The PIMD-SH method is tested with time step sizes ∆t = 1 25 , 1 50 , 1 100 , 1 200 and 1 400 with total simulation time T = 10000.…”

“…To apply methods like path-integral molecular dynamics (or dynamic extensions like ring-polymer molecular dynamics) to multilevel systems when the nonadiabatic effects cannot be neglected, a popular strategy is to use the mapping variable approach [18,19], see also the review article [2] and more recent developments in [20][21][22][23][24][25][26]. The idea is to replace the multi-level system by a single level system with higher dimension by mapping the discrete electronic states to continuous variables using uncoupled harmonic oscillators [19].…”

“…The idea is to replace the multi-level system by a single level system with higher dimension by mapping the discrete electronic states to continuous variables using uncoupled harmonic oscillators [19]. The ring polymer representation can then be applied to the mapped system [20][21][22][23]25].…”