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

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.

💡 Research Summary

This paper presents a systematic development and comparison of three trajectory‑based mixed‑quantum‑classical (MQC) approaches for simulating two‑dimensional electronic spectroscopy (2DES), a powerful technique that probes third‑order optical response functions and thus provides detailed insight into coupled electronic‑nuclear dynamics. The authors focus on (i) a pure‑state Ehrenfest method, (ii) a spin‑mapping method, and (iii) the mean classical path (MCP) approximation, evaluating each on their ability to reproduce exact linear absorption, pump‑probe, and full 2DES spectra for two Frenkel exciton models: a simple coupled dimer and the seven‑site Fenna‑Matthews‑Olson (FMO) complex.

The starting point is the standard Ehrenfest dynamics, where the electronic wavefunction evolves under a time‑dependent Schrödinger equation while the nuclear degrees of freedom follow classical equations of motion driven by the average electronic force. Because the initial light‑matter interaction creates electronic coherences (zero‑trace density‑matrix elements), these coherences must be expressed as weighted sums of pure states before propagation. The previously proposed “polar” decomposition uses four pure states that are not symmetric with respect to the two electronic states involved in the coherence. Consequently, the Ehrenfest force becomes biased toward the state with larger dipole matrix elements, leading to significant errors when the bath frequency is comparable to electronic transition frequencies or when transition dipoles are large. Moreover, the polar scheme requires 4³ = 64 trajectory branches for a third‑order response, making it computationally expensive.

The authors introduce an “equatorial” decomposition that places the pure states on the equator of the Bloch sphere: |ϕ_j⟩ = (|a⟩ + e^{i jπ/2}|b⟩)/√2 with equal complex weights ½ e^{i jπ/2} (j = 0…3). Each state contains an equal contribution from the two electronic basis states, ensuring that the Ehrenfest force is effectively the average of the ground‑ and excited‑state forces. Crucially, due to symmetry, the four pure‑state contributions collapse during the first (t₁) and third (t₃) evolution intervals of the response function, leaving only two distinct trajectories to propagate. This reduces the number of required branches from 64 to 2, a 32‑fold speed‑up, while dramatically improving accuracy for both linear and nonlinear spectra.

To address the remaining deficiency of Ehrenfest dynamics—its failure to satisfy detailed balance during the pump‑probe waiting time (t₂)—the authors replace the Ehrenfest propagation in this interval with spin‑mapping dynamics. Spin‑mapping treats the electronic subsystem as a classical spin on the SU(2) sphere, preserving quantum statistics and detailed balance. By using spin‑mapping only for t₂ and retaining the cheaper equatorial Ehrenfest (or MCP) for t₁ and t₃, they obtain a hybrid scheme that captures the correct peak intensities and line shapes in 2DES with only a modest increase in computational cost relative to pure Ehrenfest.

The simplest approach, MCP, averages the forces from the two electronic states at every time step, completely classicalizing the electronic degrees of freedom. While MCP can reproduce linear absorption reasonably well, it fails to capture coherence‑dependent features in pump‑probe and 2DES, leading to large errors in peak positions and amplitudes.

Benchmark calculations use the hierarchical equations of motion (HEOM) as an exact reference. For the dimer, the equatorial Ehrenfest reproduces the exact linear absorption and pump‑probe signals, while the hybrid spin‑mapping/Ehrenfest yields 2DES spectra that match HEOM peak intensities within a few percent. The polar decomposition shows noticeable deviations, especially in the high‑frequency bath regime, and requires the full 64‑branch propagation. For the FMO complex, similar trends are observed: the equatorial method captures the overall spectral envelope and cross‑peak structure, whereas the hybrid method restores the correct relative amplitudes of diagonal and off‑diagonal peaks. MCP, by contrast, misplaces cross‑peaks and underestimates their strengths.

The authors discuss practical implications for ab‑initio simulations. The equatorial decomposition needs only a single electronic wavefunction per trajectory, making it compatible with on‑the‑fly TDDFT or other electronic structure methods without additional mapping variables. The spin‑mapping step introduces extra phase‑space variables but is embarrassingly parallel and can be efficiently sampled. Consequently, the hybrid scheme offers a viable route to high‑accuracy 2DES simulations of realistic chromophore assemblies at a fraction of the cost of fully linearized methods such as PLDM or spin‑PLDM, which typically require 10⁶–10⁹ trajectories.

In summary, the paper demonstrates that (1) an equatorial pure‑state decomposition dramatically reduces the computational burden of Ehrenfest‑based response calculations while improving accuracy, (2) spin‑mapping applied only during the waiting time restores detailed balance and yields quantitatively correct 2DES spectra, and (3) the mean classical path approximation, although cheapest, is insufficient for quantitative spectroscopy. The hybrid equatorial‑Ehrenfest/spin‑mapping approach emerges as the most promising balance of cost and fidelity, opening the door to routine, first‑principles simulation of multidimensional electronic spectroscopies for complex photosynthetic and material systems.