Analysis of divergent dynamics of exactly factorized electron-nuclear wavefunctions

TL;DR

This work addresses numerical instabilities in the Exact Factorization (XF) treatment of electron-nuclear dynamics that arise when derivative couplings involve division by the nuclear density. Using a minimal two-state photodissociation model, the authors analyze the XF formalism and quantum-trajectory (QT) representation, defining XF electronic coefficients $C_i(y,t)$ and showing that as the nuclear wavepackets diverge, these coefficients exhibit step-like behavior with near-discontinuous gradients. They derive analytical estimates for the location and magnitude of extrema in $|C_i|$ near the median $y_m$, and demonstrate that derivative terms can become nearly singular, with the severity scaling as $t^3$ in the moving frame; this persists in both BO and atomic representations. The paper concludes that switching to state-specific trajectory descriptions after divergence could mitigate these XF instabilities and provides analytic insight to guide future XF-method development for non-adiabatic dynamics. This work advances understanding of XF numerical challenges and informs practical strategies for applying XF to larger, more complex systems.

Abstract

The Exact Factorization (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, 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 analyzed analytically. We demonstrate that the problem arises when the factorized 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 analyze 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 behavior 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.

Figures (7)

• Figure 1: A minimal model of photodissociation dynamics based on two electronic states. (a) A fraction of the nuclear wavepacket, $\psi_1(0)$, in the ground electronic state (red parabola) is instantaneously transferred into the excited electronic state (blue linear function) of dissociative character. With time this wavepacket, $\psi_2$, diverges from $\psi_1$ in the nuclear coordinate. (b) The probability densities of the Born-Huang nuclear wavepackets $\psi_1$ (red solid line) and $\psi_2$ (blue solid line) add up to the XF nuclear wavefunction, $|\psi|^2=\sum_i|\psi_i|^2$, defining the expansion amplitudes, $|C_1|$ (red dash) and $|C_2|$ (blue dash) of the XF complement to $\psi$ in the electron-nuclear space. For diverging wavepackets, $|C_i|$ behave as step-functions near the median point of equal $|\psi_i|^2$ values.
• Figure 2: The quantum trajectories and associated electronic populations for the photodissociation model. Panels (a) and (c) show the QTs (solid lines, Eq. (\ref{['eq:rhoc']})) for the exact and frozen wavepackets, respectively. The red dashed lines are the centers of the ground and excited state Gaussians (note the ground state center remains at zero for all time). The green dot dash line marks the position of equal density, $y_m(t)$. Panels (b) and (d) show the populations of the electronic ground state along each trajectory, with the colors of the solid lines corresponding to trajectories plotted in (a) and (c) correspondingly. The black open circles indicate the point where the dissociating trajectories cross $y_m(t)$.
• Figure 3: The maximum time-derivative of $|C_1|$ in the Eulerian and Lagrangian frames of reference computed for $\eta=1$ and $\eta=2$ using $M=100$ and $k=20$ a.u. (a) Position with respect to the median, $y_{max}-y_m$, and (b) the largest amplitude of the extrema of the time-derivative, $\partial|C_1|/\partial t$ and of $d|C_1|/dt$ as functions of $q_t$ for the Eulerian and Lagrangian frames of reference, respectively.
• Figure 4: Time-derivative of the ground state coefficient amplitude in (a-c) the stationary Eulerian and (d-f) moving Lagrangian frames with $\delta$ indicating the relative position to the median point. For the moving frame, in the $\eta=1$ case, the point of 0 amplitude change is always at the median; otherwise its relative position changes. The size of the derivatives increases as $q_t\Bar{p}_t$, or cubically in time as given by Eqs (\ref{['eq:analyticpdapdt']}) and (\ref{['eq:analyticdadt']}).
• Figure 5: XF coefficient amplitudes $|C_i|^2$ for $i=\{L,R\}$ as green and blue dashed lines. The Born-Huang ground and excited state wavepackets are shown as red and black solid lines respectively. Snapshots of the nuclear density at different separations and the corresponding electronic coefficients in the atomic orbital basis, Eq. (\ref{['eq:LocalizedPhi']}). We see the presence of an interference pattern near the median point for FWPs. Initial conditions given in Table \ref{['tab:ModelParamsXFPD']}.