On the Numerical Solution of the Exact Factorization Equations

Abstract

The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schrödinger 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 Schrödinger 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.

14 pages, 7 figures

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• Density functional theory beyond the Born-Oppenheimer approximation: Exact mapping onto an electronically non-interacting Kohn-Sham molecule
• Nucleus-electron correlation revising molecular bonding fingerprints from the exact wavefunction factorization
• Characterization of the continuous transition from atomic to molecular shape in the three-body Coulomb system
• Electron Transfer, Diabatic Couplings and Vibronic Energy Gaps in a Phase Space Electronic Structure Framework