Using a multistate mapping approach to surface hopping to predict the ultrafast electron diffraction signal of gas-phase cyclobutanone.

Using the recently developed multistate mapping approach to surface hopping (multistate MASH) method combined with SA(3)-CASSCF(12,12)/aug-cc-pVDZ electronic structure calculations, the gas-phase isotropic ultrafast electron diffraction (UED) of cyclobutanone is predicted and analyzed. After excitation into the n-3s Rydberg state (S2), cyclobutanone can relax through two S2/S1 conical intersections, one characterized by compression of the CO bond and the other by dissociation of the α-CC bond. Subsequent transfer into the ground state (S0) is then achieved via two additional S1/S0 conical intersections that lead to three reaction pathways: α ring-opening, ethene/ketene production, and CO liberation. The isotropic gas-phase UED signal is predicted from the multistate MASH simulations, allowing for a direct comparison to the experimental data. This work, which is a contribution to the cyclobutanone prediction challenge, facilitates the identification of the main photoproducts in the UED signal and thereby emphasizes the importance of dynamics simulations for the interpretation of ultrafast experiments.

Exciton dynamics from the mapping approach to surface hopping: comparison with Förster and Redfield theories.

We compare the recently introduced multi-state mapping approach to surface hopping (MASH) with the Förster and Redfield theories of excitation energy transfer. Whereas Förster theory relies on weak coupling between chromophores, and Redfield theory assumes the electronic excitations to be weakly coupled to fast chromophore vibrations, MASH is free from any perturbative or Markovian approximations. We illustrate this with an example application to the rate of energy transfer in a Frenkel-exciton dimer, showing that MASH interpolates correctly between the opposing regimes in which the Förster and Redfield results are reliable. We then compare the three methods for a realistic model of the Fenna-Matthews-Olson complex with a structured vibrational spectral density and static disorder in the excitation energies. In this case there are no exact results for comparison so we use MASH to assess the validity of Förster and Redfield theories. We find that Förster theory is the more accurate of the two on the picosecond timescale, as has been shown previously for a simpler model of this particular light-harvesting complex. We also explore various ways to sample the initial electronic state in MASH and find that they all give very similar results for exciton dynamics.

A multi-state mapping approach to surface hopping.

We describe a multiple electronic state adaptation of the mapping approach to surface hopping introduced recently by Mannouch and Richardson [J. Chem. Phys. 158, 104111 (2023)]. Our modification treats populations and coherences on an equal footing and is guaranteed to give populations in any electronic basis that tend to the correct quantum-classical equilibrium values in the long-time limit (assuming ergodicity). We demonstrate its accuracy by comparison with exact benchmark results for three- and seven-state models of the Fenna-Matthews-Olson complex, obtaining electronic populations and coherences that are significantly more accurate than those of fewest switches surface hopping and at least as good as those of any other semiclassical method we are aware of. Since these results were obtained by adapting the scheme of Mannouch and Richardson, we go on to compare our results with theirs for a variety of problems with two electronic states. We find that their method is sometimes more accurate, especially in the Marcus inverted regime. However, in other situations, the accuracies are comparable, and since our scheme can be used with multiple electronic states it can be applied to a wider variety of electronically nonadiabatic systems.

How Quantum is the Resonance Behavior in Vibrational Polariton Chemistry?

Recent experiments in polariton chemistry have demonstrated that reaction rates can be modified by vibrational strong coupling to an optical cavity mode. Importantly, this modification occurs only when the frequency of the cavity mode is tuned to closely match a molecular vibrational frequency. This sharp resonance behavior has proved to be difficult to capture theoretically. Only recently did Lindoy et al. [ Nat. Commun. 2023, 14, 2733] report the first instance of a sharp resonant effect in the cavity-modified rate simulated in a model system using exact quantum dynamics. We investigate the same model system with a different method, ring-polymer molecular dynamics (RPMD), which captures quantum statistics but treats dynamics classically. We find that RPMD does not reproduce this sharp resonant feature at the well frequency, and we discuss the implications of this finding for future studies of vibrational polariton chemistry.

Which Algorithm Best Propagates the Meyer-Miller-Stock-Thoss Mapping Hamiltonian for Non-Adiabatic Dynamics?

A common strategy to simulate mixed quantum-classical dynamics is by propagating classical trajectories with mapping variables, often using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian or the related spin-mapping approach. When mapping the quantum subsystem, the coupled dynamics reduce to a set of equations of motion to integrate. Several numerical algorithms have been proposed, but a thorough performance comparison appears to be lacking. Here, we compare three time-propagation algorithms for the MMST Hamiltonian: the Momentum Integral (MInt) (J. Chem. Phys., 2018, 148, 102326), the Split-Liouvillian (SL) (Chem. Phys., 2017, 482, 124-134), and the algorithm in J. Chem. Phys., 2012, 136, 084101 that we refer to as the Degenerate Eigenvalue (DE) algorithm due to the approximation required during derivation. We analyze the accuracy of individual trajectories, correlation functions, energy conservation, symplecticity, Liouville's theorem, and the computational cost. We find that the MInt algorithm is the only rigorously symplectic algorithm. However, comparable accuracy at a lower computational cost can be obtained with the SL algorithm. The approximation implicitly made within the DE algorithm conserves energy poorly, even for small timesteps, and thus leads to slightly different results. These results should guide future mapping-variable simulations.

On detailed balance in nonadiabatic dynamics: From spin spheres to equilibrium ellipsoids.

Trajectory-based methods that propagate classical nuclei on multiple quantum electronic states are often used to simulate nonadiabatic processes in the condensed phase. A long-standing problem of these methods is their lack of detailed balance, meaning that they do not conserve the equilibrium distribution. In this article, we investigate ideas for restoring detailed balance in mixed quantum-classical systems by tailoring the previously proposed spin-mapping approach to thermal equilibrium. We find that adapting the spin magnitude can recover the correct long-time populations but is insufficient to conserve the full equilibrium distribution. The latter can however be achieved by a more flexible mapping of the spin onto an ellipsoid, which is constructed to fulfill detailed balance for arbitrary potentials. This ellipsoid approach solves the problem of negative populations that has plagued previous mapping approaches and can therefore be applied also to strongly asymmetric and anharmonic systems. Because it conserves the thermal distribution, the method can also exploit efficient sampling schemes used in standard molecular dynamics, which drastically reduces the number of trajectories needed for convergence. The dynamics does however still have mean-field character, as is observed most clearly by evaluating reaction rates in the golden-rule limit. This implies that although the ellipsoid mapping provides a rigorous framework, further work is required to find an accurate classical-trajectory approximation that captures more properties of the true quantum dynamics.

Explaining the Efficiency of Photosynthesis: Quantum Uncertainty or Classical Vibrations?

Photosynthetic organisms are known to use a mechanism of vibrationally assisted exciton energy transfer to efficiently harvest energy from light. The importance of quantum effects in this mechanism is a long-standing topic of debate, which has traditionally focused on the role of excitonic coherences. Here, we address another recent claim: that the efficient energy transfer in the Fenna-Matthews-Olson complex relies on nuclear quantum uncertainty and would not function if the vibrations were classical. We present a counter-example to this claim, showing by trajectory-based simulations that a description in terms of quantum electrons and classical nuclei is indeed sufficient to describe the funneling of energy to the reaction center. We analyze and compare these findings to previous classical-nuclear approximations that predicted the absence of an energy funnel and conclude that the key difference and the reason for the discrepancy is the ability of the trajectories to properly account for Newton's third law.

Spin-Mapping Methods for Simulating Ultrafast Nonadiabatic Dynamics.

Many chemical reactions exhibit nonadiabatic effects as a consequence of coupling between electronic states and/or interaction with light. While a fully quantum description of nonadiabatic reactions is unfeasible for most realistic molecules, a more computationally tractable approach is to combine a classical description of the nuclei with a quantum description of the electronic states. Combining the formalisms of quantum and classical dynamics is however a difficult problem for which standard methods (such as Ehrenfest dynamics and surface hopping) may be insufficient. In this article, we review a new trajectory-based approach developed in our group that is able to describe nonadiabatic dynamics with a higher accuracy than previous approaches but for a similar level of computational effort. This method treats the electronic states with a phase-space representation for discrete-level systems, which in the two-level case is analogous to a spin-½. We point out the key features of the method and demonstrate its use in a variety of applications, including ultrafast transfer through conical intersections, damped coherent excitation under coupling to a strong light field, and nonlinear spectroscopy of light-harvesting complexes.

Quantum Entanglement from Classical Trajectories.

A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.

Generalized spin mapping for quantum-classical dynamics.

We recently derived a spin-mapping approach for treating the nonadiabatic dynamics of a two-level system in a classical environment [J. E. Runeson and J. O. Richardson, J. Chem. Phys. 151, 044119 (2019)] based on the well-known quantum equivalence between a two-level system and a spin-1/2 particle. In the present paper, we generalize this method to describe the dynamics of N-level systems. This is done via a mapping to a classical phase space that preserves the SU(N)-symmetry of the original quantum problem. The theory reproduces the standard Meyer-Miller-Stock-Thoss Hamiltonian without invoking an extended phase space, and we thus avoid leakage from the physical subspace. In contrast to the standard derivation of this Hamiltonian, the generalized spin mapping leads to an N-dependent value of the zero-point energy parameter that is uniquely determined by the Casimir invariant of the N-level system. Based on this mapping, we derive a simple way to approximate correlation functions in complex nonadiabatic molecular systems via classical trajectories and present benchmark calculations on the seven-state Fenna-Matthews-Olson light-harvesting complex. The results are significantly more accurate than conventional Ehrenfest dynamics, at a comparable computational cost, and can compete in accuracy with other state-of-the-art mapping approaches.

Spin-mapping approach for nonadiabatic molecular dynamics.

We propose a trajectory-based method for simulating nonadiabatic dynamics in molecular systems with two coupled electronic states. Employing a quantum-mechanically exact mapping of the two-level problem to a spin-12 coherent state, we use the Stratonovich-Weyl transform to construct a classical phase space of a spin vector constrained to a spherical surface whose radius is consistent with the quantum magnitude of the spin. In contrast with the singly excited harmonic oscillator basis used in Meyer-Miller-Stock-Thoss (MMST) mapping, the theory requires no additional projection operators onto the space of physical states. When treated under a quasiclassical approximation, we show that the resulting dynamics are equivalent to those generated by the MMST Hamiltonian. What differs is the value of the zero-point energy parameter as well as the initial distribution and the measurement operators used in constructing correlation functions. For various spin-boson models, the results of the method are seen to be a significant improvement compared to both standard Ehrenfest dynamics and linearized semiclassical MMST mapping, without adding any computational complexity.