The second-order Ehrenfest method

Abstract

This article describes the Ehrenfest method and our second-order implementation (with approximate gradient and Hessian) within a CASSCF formalism. We demonstrate that the second order implementation with the predictor-corrector integration method improves the accuracy of the simulation significantly in terms of energy conservation. Although the method is general and can be used to study any coupled electron-nuclear dynamics, we apply it to investigate charge migration upon ionization of small organic molecules, focusing on benzene cation. Using this approach, we can study the evolution of a non-stationary electronic wavefunction for fixed atomic nuclei, and where the nuclei are allowed to move, to investigate the interplay between them for the first time. Analysis methods for the interpretation of the electronic and nuclear dynamics are suggested: we monitor the electronic dynamics by calculating the spin density of the system as a function of time.

Electron dynamics upon ionization: Control of the timescale through chemical substitution and effect of nuclear motion

“…16 Here, we only provide a summary and illustrate the method with a two-state system (as we will consider the two lowestenergy states populated in the molecules studied).…”

“…(1) using product ansatz (2) and taking the classical limit for the nuclei gives the following equation for the quantum mechanical treatment of the electron dynamics: 16 i ∂Ψ(r,t; R(t)) ∂t = H e (r; R(t)) · Ψ(r,t; R(t)),…”

“…The approximations made are detailed in our previous article. 16 One limitation of the Ehrenfest method is the lack of electronic decoherence: the populated electronic eigenstates share the same nuclear geometry because of the ansatz for the total wavefunction (Eq. (2)).…”

“…A theoretical study by Nest and Ulusoy 14 suggests that the effect of nuclear motion is very small during the first ten or so femtoseconds, as it takes time for the nuclei to move sufficiently to influence the electronic superposition. We have previously studied molecules 15,16 where electron dynamics is affected by the nuclear motion after only a few fs and others 15,17 when it seems to survive for several tens of fs. One of the purposes of this article is to understand further the extent to which neglecting nuclear motion is a reasonable approximation.…”

“…24 The justification for the fixed-nuclei approximation-different timescales for electron and nuclear dynamics-therefore does not hold in regions of the potential energy surfaces near CIs. 25 Using our complete active space self-consistent field (CASSCF) implementation of the Ehrenfest method, 16 we can study the evolution of a non-stationary electronic wavefunction for fixed nuclei, and where the nuclei are allowed to move, to investigate the differences. 15,17 We can therefore study the influence of the nuclear motion on the electron dynamics and see to what extent the fixed-nuclei approximation is valid.…”