Analysis of divergent dynamics of exactly factorised electron-nuclear wavefunctions

The exact factorisation (XF) of molecular wavefunctions can be viewed as an ‘electronic wavepacket’ framework for quantum dynamics. It is an appealing alternative to the conventional non-adiabatic dynamics, unfolding in the space of coupled electronic eigenstates. However, the implementation of the non-linear XF equations for general systems presents a formidable challenge: the XF counterparts to the non-adiabatic coupling involve division by the nuclear probability density, which leads to severe numerical instabilities in the low-density regions of space. In case of the non-adiabatic dynamics, the effect of coupling is relatively smooth, but this theoretical framework becomes impractical when numerous electronic states are involved. In this paper, the origin of the XF-specific challenge is analysed analytically. We demonstrate that the problem arises when the factorised wavefunction diverges, even without the explicit coupling of the Born–Huang electronic states used to describe the molecular wavefunction. Using a ‘minimal’ model of the photodissociation, we derive expressions for the XF dynamics and locate the source of the XF instability. We analyse the dependence of this instability on the nuclear wavefunction bifurcation in the stationary and moving frames of reference, the latter associated with the quantum trajectory ensemble describing the nuclear XF wavepacket in a compact form. We show that the near-singular behaviour persists in the moving frame and in the atomic basis representation of the electronic wavefunction. This model and insight into the root of the XF implementation challenge will help to address the issue, leading to further development of the XF methods.