Phenomenological Ehrenfest Dynamics with Topological and Geometric Phase Effects and the curious case of Elliptical intersection

TL;DR

The study addresses how geometric-phase (GP) effects influence nonadiabatic molecular dynamics by embedding Berry-curvature corrections into a phenomenological Ehrenfest framework for a two-level system. It introduces a unified model capable of representing conical, avoided, and elliptic intersections, and implements a pre-looping trajectory initialization to encode GP memory in the initial phase, along with analytic Berry-curvature force corrections. The approach yields results consistent with theoretical GP predictions, reveals distinct force landscapes across crossing types, and shows that elliptic intersections support a tunable, path-invariant Berry phase distinct from the CI’s $\'\gamma=\pi\'. The framework offers a versatile tool for simulating quantum-classical dynamics where GP effects are pronounced, with potential implications for spectroscopy design and degenerate-material phenomena.

Abstract

We present a comprehensive computational framework for simulating nonadiabatic molecular dynamics with explicit inclusion of geometric phase (GP) effects. Our approach is based on a generalized two-level Hamiltonian model that can represent various electronic state crossings - conical intersections, avoided crossings, and elliptic intersections - through appropriate parameterization. We introduce a novel prelooping trajectory initialization scheme, allowing us to encode the memory as an initial phase accumulated due to the adiabatic evolution over the potential energy surface. This is a unified framework to handle different types of level crossings by incorporating Berry curvature-based force corrections to Ehrenfest dynamics, ensuring accurate representation of topological effects. For conical intersections, our method incorporates the theoretically expected phase pi, while for elliptic intersections, it yields a parametrically tunable but loop radius (energy) independent phase different from pi. We also include an eccentricity parameter (e) in the diabatic coupling to model more realistic molecular systems. Numerical simulations demonstrate the consistency of our approach with theoretical predictions for mixing of states and inhibition from mixing due to geometric phase effects. This framework provides a valuable tool for studying quantum-classical interactions in molecular systems where geometric phase effects play a significant role. The elliptical intersection and geometric phase effect opens avenue for the design and discovery of degenerate materials. It produces a fresh look to help develop a new kind of spectroscopy and potential qubit applications. This simple Hamiltonian reveals a pathological phase protection effect E = kr, where k is real, that has great utility in a new spectroscopy design.

Figures (7)

• Figure 1: Visualization of quantum-classical force components in a two-level system with tunable intersection. Red transparent surfaces show the lower and upper adiabatic potential energy surfaces (PES) as functions of nuclear coordinates $(x, y)$, computed from the model Hamiltonian. Blue surfaces represent the Berry curvature (clipped for visibility) for the lower and upper adiabatic states, highlighting regions of strong geometric phase effects. Light green and dark green lines indicate the real parts of the $x$- and $y$-components of the nonadiabatic coupling (NAC) vector, respectively, computed numerically on a fine grid near the intersection. This comprehensive visualization illustrates the interplay between mean-field, geometric, and nonadiabatic forces in quantum molecular dynamics near electronic state crossings.
• Figure 2: Visualization of the different types of electronic state crossings for $a=s=1$ and $e=0$. The adiabatic surfaces of our model Hamiltonian showing how different choices of $Z$ lead to: (a) conical intersection, (b) avoided crossing, and (c) elliptic intersection. The red and blue surfaces represent the upper and lower adiabatic energy surfaces, respectively. Note that while the elliptic intersection appears visually similar to the conical intersection near the crossing point, it exhibits stronger curvature with steeper gradients, spanning a wider energy range (-3 to 3) compared to the conical intersection's narrower range (-2 to 2).
• Figure 3: Berry phase variation with loop radius for different energy gaps $E$ in our model $2X2$ Hamiltonian, showing characteristic behavior around the avoided crossings. Smaller $E$ values produce sharper transitions in the phase, while larger $E$ values lead to more gradual changes, reflecting the topographical nature of the phase accumulation in quantum systems. The dots represents the Berry's phase for the avoided crossings at the points $E=r$, revealing the phase due to elliptical intersection.
• Figure 4: Trajectories evolving over simulation time as Gaussian wavepackets in the adiabatic basis. The trajectories are plotted with the adiabatic potential energy surfaces in represents the background energy landscape. The visualization shows three distinct trajectories (green, purple, and yellow) with their respective starting $(\bullet)$ and ending $(\times)$ points marked. The blue and red surfaces represent the upper and lower adiabatic energy surfaces, respectively. Simulation performed with 500 total trajectories - 250 regular trajectories and 250 are pre-looping.
• Figure 5: The panels represent: (a) conical intersection (E=0.0), (b) avoided crossing (E=0.05), and (c) elliptic intersection (E=func). The parameters $a=s=1$ and $e=0$ for all the three crossing types. Each panel shows the time evolution of adiabatic state populations, with solid lines representing prelooping trajectories and dashed lines representing regular trajectories. Blue lines correspond to the lower adiabatic state and red lines to the upper adiabatic state.