Two-dimensional electronic spectra from trajectory-based dynamics: pure-state Ehrenfest, spin-mapping, and mean classical path approaches

TL;DR

This study addresses the challenge of computing accurate 2DES spectra from trajectory-based nonadiabatic dynamics. It introduces an equatorial pure-state Ehrenfest decomposition that collapses two of the three pure-state sums, achieving up to ~32× cost reductions, and demonstrates that the resulting spectra closely match exact benchmarks for biexciton and FMO models; spin mapping is added to ameliorate detailed-balance issues during the pump-probe delay, with some trade-offs in cost and accuracy. The authors compare these approaches to the mean classical path approximation, finding near-equivalence in many cases and highlighting the MCP’s appeal for large-scale, ab initio-capable simulations. While Ehrenfest-based methods retain known limitations in detailed balance and zero-point energy leakage, the work outlines practical routes (MASH, polaron transformations, machine-learned potentials) to extend their applicability to realistic, atomistic systems. Overall, the equatorial Ehrenfest framework offers a practical path toward efficient, interpretable 2DES simulations with trajectory-based dynamics.

Abstract

Two-dimensional electronic spectroscopy (2DES) provides a detailed picture of electronically nonadiabatic dynamics that can be interpreted with the aid of simulations. Here, we develop and contrast trajectory-based nonadiabatic dynamics approaches for simulating 2DES spectra. First, we argue that an improved pure-state Ehrenfest approach can be constructed by decomposing the initial coherence into a sum of equatorial pure states that contain equal contributions from the states in the coherence. We then use this framework to show how one can obtain a more accurate, but computationally more expensive, approximation to the third-order 2DES response function by replacing Ehrenfest dynamics with spin mapping during the pump-probe delay time. We end by comparing and contrasting the accuracy of these methods and the simpler mean classical path approximation in reproducing the exact linear, pump-probe, and 2DES spectra of two Frenkel exciton models: a coupled dimer system and the Fenna-Matthews-Olson (FMO) complex.

Figures (12)

• Figure 1: Feynman diagrams for the pathways (nonlinear response functions) that survive the rotating wave approximation. Right-pointing arrows correspond to the $e^{-i\omega t + i\mathbf{k} \cdot \mathbf{r}}$ component and left-pointing arrows to the $e^{+i\omega t - i\mathbf{k} \cdot \mathbf{r}}$ component of the electric field. Arrows pointing towards the center of the diagram symbolize an excitation, while arrows pointing away symbolize a de-excitation. The colors denote the ground ($|g\rangle$, black straight), singly excited ($|s\rangle$, red zigzag) and doubly excited ($|d\rangle$, blue wavy) manifolds of the bra and ket states.
• Figure 2: Bloch sphere visualization of (a) polar and (b) equatorial pure state decompositions. The states $|a\rangle$ and $|b\rangle$ are at the north and south poles in both cases. The 4 pure states in each decomposition, along with $|a\rangle$ and $|b\rangle$, are drawn as independently normalized. Each pure state is labeled by its weight in Eq. \ref{['eq:def-pure-state-decomp-long']} or \ref{['eq:def-pure-state-decomp-eq-weights']}, with the factors of 1/2 in Eq. \ref{['eq:def-pure-state-decomp-eq-weights']} omitted for clarity.
• Figure 3: Comparison of polar, equatorial, and spin mapping 2DES spectra for the biexciton model with numerically exact HEOM results, as a function of the delay time $t_2$. The three quantum-classical spectra were computed with 96,000 trajectories. All spectra are normalized to a maximum value of 1 at $t_2=0$.
• Figure 4: Comparison of polar, equatorial, and spin mapping diagonal cuts (top) and pump-probe spectra (bottom) for the biexciton model with numerically exact HEOM results, as a function of the delay time $t_{2}$. All spectra are normalized to a maximum value of 1 at $t_2=0$.
• Figure 5: Linear absorption spectra of the biexciton model obtained using the polar and equatorial approaches, compared with the numerically exact HEOM benchmark. The spin-mapping approach is identical to the equatorial approach for the linear spectrum and therefore not shown as a separate curve.