# On the Numerical Solution of the Exact Factorization Equations

## TL;DR

This paper demonstrates a self-consistent numerical solution to the exact factorization equations, revealing their stability properties and implications for non-adiabatic molecular dynamics simulations.

## Contribution

It introduces a novel approach to solve the EF equations self-consistently, addressing stability issues due to their non-Hermitian form.

## Key findings

- Stable propagation achieved in model systems
- Identification of non-Hermitian Hamiltonian effects
- Insights into non-adiabatic dynamics modeling

## Abstract

The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schr\"odinger equation as two coupled equations, one for the nuclear wavefunction and one for the conditional electronic wavefunction. The potentials appearing in these equations have provided insight into non-adiabatic processes, and new practical non-adiabatic dynamics methods have been formulated starting from these equations. Here we provide a first demonstration of a self-consistent solution of the exact equations, with a preliminary analysis of their stability and convergence properties. The equations have an unprecedented mathematical form, involving a non-Hermitian Hamiltonian, and so the usual numerical methods for time-dependent Schr\"odinger fail when applied in a straightforward way to the EF equations. We find an approach that enables stable propagation long enough to witness non-adiabatic behavior in a model system before non-trivial instabilities take over. Implications for the development and analysis of EF-based methods are discussed.