Polaron formation: Ehrenfest dynamics vs. exact results

TL;DR

This study compares semiclassical Ehrenfest dynamics with exact quantum calculations to understand polaron formation, revealing that Ehrenfest dynamics fails to accurately capture the timescale of the process due to its classical treatment of nuclear motion.

Contribution

The paper provides a detailed comparison between Ehrenfest semiclassical dynamics and exact quantum methods for modeling polaron formation, highlighting the limitations of the semiclassical approach.

Findings

Both methods yield similar long-term probabilities for electron trapping and escape.

Ehrenfest dynamics does not accurately predict the timescale of polaron formation.

Quantum calculations fully account for correlations between electronic and vibrational states.

Abstract

We use a 1-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bon\v{c}a and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron in a polaron cage and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct timescale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the otherwise-distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.