# The second-order Ehrenfest method

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## 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.

## Keywords

Ehrenfest method CASSCF Coupled electron–nuclear dynamics Charge migration Charge transfer## Notes

### Acknowledgments

This work was supported by UK-EPSRC Grant EP/I032517/1. All calculations were run using the Imperial College High Performance Computing service. The original SA CP-MCSCF programs were written by Thom Vreven. The work on the Ehrenfest programs was initiated by Patricia Hunt (see the supplementary information of reference [9]).

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