Multiconfigurational Ehrenfest approach to quantum coherent dynamics in large molecular systems

Simulations of photoinduced processes with the exact factorization: state of the art and perspectives

This perspective offers an overview of the applications of the exact factorization of the electron-nuclear wavefunction to the domain of theoretical photochemistry, where the aim is to gain insights into the ultrafast dynamics of molecular systems via simulations of their excited-state dynamics beyond the Born-Oppenheimer approximation. The exact factorization offers an alternative viewpoint to the Born-Huang representation for the interpretation of dynamical processes involving the electronic ground and excited states as well as their coupling through the nuclear motion. Therefore, the formalism has been used to derive algorithms for quantum molecular-dynamics simulations where the nuclear motion is treated using trajectories and the electrons are treated quantum mechanically. These algorithms have the characteristic features of being based on coupled and on auxiliary trajectories, and have shown excellent performance in describing a variety of excited-state processes, as this perspective illustrates. We conclude with a discussion on the authors' point of view on the future of the exact factorization.

Time-Dependent Multireference Coupled-Cluster Method (TDMRCCM) for the Bath-Dynamics in System-Bath Approach to Nonadiabatic Dynamics

The time-dependent multireference coupled-cluster method (TDMRCCM) fits well in the scheme of the system-bath separation used to study the nonadiabatic dynamics. In TDMRCCM, a projection operator is defined as one that projects the full Hilbert space onto the space spanned by the collection of system degrees of freedom, called the model space. The inverse of this projection operator is a wave operator that acts on the model space and takes its projection back to the full wave function. This wave operator is defined as an exponential of the excitation terms and, hence, can be expanded into a Taylor series, which we have truncated in this work at the second-order of excitations. The attraction of using TDMRCCM for describing the bath dynamics is due to the exponential nature of the ansatz used in the method, which makes it possible for the higher-order excitations to be absorbed by the lower-order terms, even upon truncating the series. This improves the accuracy of the numerical calculations using TDMRCCM as an approximation for the bath-mode dynamics in the system-bath framework for nonadiabatic dynamics. We present the theoretical details of TDMRCCM and the numerical results for implementing this method to study the dynamics in the butatriene cation.

Ultrafast electron diffraction of photoexcited gas-phase cyclobutanone predicted by ab initio multiple cloning simulations

We present the result of our calculations of ultrafast electron diffraction (UED) for cyclobutanone excited into the S2 electronic state, which is based on the non-adiabatic dynamics simulations with the Ab Initio Multiple Cloning (AIMC) method with the electronic structure calculated at the SA(3)-CASSCF(12,12)/aug-cc-pVDZ level of theory. The key features in the UED pattern were identified, which can be used to distinguish between the reaction pathways observed in the AIMC dynamics, although there is a significant overlap between representative signals due to the structural similarity of the products. The calculated UED pattern can be compared with the experiment.

Prediction of fluorescence quantum yields using the extended thawed Gaussian approximation

Spontaneous emission and internal conversion rates are calculated within harmonic approximations and compared to the results obtained within the semi-classical extended thawed Gaussian approximation (ETGA). This is the first application of the ETGA in the calculation of internal conversion and emission rates for real molecular systems, namely, formaldehyde, fluorobenzene, azulene, and a dicyano-squaraine dye. The viability of the models as black-box tools for prediction of spontaneous emission and internal conversion rates is assessed. All calculations were done using a consistent protocol in order to investigate how different methods perform without previous experimental knowledge using density functional theory (DFT) and time-dependent DFT (TD-DFT) with B3LYP, PBE0, ωB97XD, and CAM-B3LYP functionals. Contrasting the results with experimental data shows that there are further improvements required before theoretical predictions of emission and internal conversion rates can be used as reliable indicators for the photo-luminescence properties of molecules. We find that the ETGA performs rather similar to the vertical harmonical model. Including anharmonicities in the calculation of internal conversion rates has a moderate effect on the quantitative results in the studied systems. The emission rates are fairly stable with respect to computational parameters, but the internal conversion rate reveals itself to be highly dependent on the choice of the spectral line shape function, particularly the width of the Lorentzian function, associated with homogeneous broadening.

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.

Criteria for the Classification of System and Bath Variables in Nonadiabatic Dynamics: Application to Pyrazine

The present work deals with the system-bath separation of the vibrational modes involved in a nonadiabatic system. System modes are the strongly interacting modes which dominate the overall dynamics and, hence, need a near-exact treatment. Bath modes have relatively weaker couplings and can be approximately treated. Thus, the exponential bottleneck involved in computations is controlled by the size of the system-subspace. The aim of this work is to put forward a set of criteria that provides clear guidelines for choosing the system degrees of freedom. The extent of wave packet dephasing caused by repeated crossings across the surface of curve-crossing is the basis for distinction between system and bath modes. The mechanisms of the wave packet dephasing and the criteria are discussed in detail. Numerically converged results presented for the 24-mode pyrazine and 3-mode spin-boson model validate the efficiency of these criteria.

A mapping approach to surface hopping

We present a nonadiabatic classical-trajectory approach that offers the best of both worlds between fewest-switches surface hopping (FSSH) and quasiclassical mapping dynamics. This mapping approach to surface hopping (MASH) propagates the nuclei on the active adiabatic potential-energy surface, such as in FSSH. However, unlike in FSSH, transitions between active surfaces are deterministic and occur when the electronic mapping variables evolve between specified regions of the electronic phase space. This guarantees internal consistency between the active surface and the electronic degrees of freedom throughout the dynamics. MASH is rigorously derivable from exact quantum mechanics as a limit of the quantum-classical Liouville equation (QCLE), leading to a unique prescription for momentum rescaling and frustrated hops. Hence, a quantum-jump procedure can, in principle, be used to systematically converge the accuracy of the results to that of the QCLE. This jump procedure also provides a rigorous framework for deriving approximate decoherence corrections similar to those proposed for FSSH. We apply MASH to simulate the nonadiabatic dynamics in various model systems and show that it consistently produces more accurate results than FSSH at a comparable computational cost.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Ultrafast coherent photoexcited dynamics in a trimeric dendrimer probed by X-ray stimulated-Raman signals

The photoinduced ultrafast coherent inter-chromophore energy redistribution in a triarylamine trimer is explored using nonadiabatic excited state molecular dynamics followed by simulations of X-ray Raman signals. The nitrogencentered system ensures strong interchromophore interactions and, thus, the presence of coherences. Nevertheless, the multitude of non-deterministic photoinduced pathways during the ultrafast inter-branch migration of the excitation results in random confinement on some branches and, therefore, spatial exciton scrambling and loss of phase information at long times. We show that the vibronic coherence dynamics evolving into the incoherent scrambling mechanism on ultrafast 50 fs timescale, is accurately probed by the TRUECARS X-ray stimulated Raman signal. In combination with previous results, where the technique has revealed long-lived coherences in a rigid heterodimer, the signal is most valuable for detecting ultrafast molecular coherences or their absence. We demonstrate that X-ray Raman spectroscopy is a useful tool in the chemical design of functional molecular building blocks.

The photoinduced ultrafast coherent inter-chromophore energy redistribution in a triarylamine trimer is explored using nonadiabatic excited state molecular dynamics followed by simulations of X-ray Raman signals.

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.

Multi-configuration electron-nuclear dynamics: An open-shell approach

The multi-configuration electron-nuclear dynamics for open shell systems with a spin-unrestricted formalism is described. The mean fields are evaluated using second-order reduced density matrices for electronic and nuclear degrees of freedom. Applications to light-element diatomics including equilibrium geometries, electronic energies, dipole moments, and absorption spectra are presented. The von Neumann entropies for different spin states of a LiH molecule are compared.

Simulating Vibronic Spectra without Born-Oppenheimer Surfaces

We show how linear vibronic spectra in molecular systems can be simulated efficiently using first-principles approaches without relying on the explicit use of multiple Born-Oppenheimer potential energy surfaces. We demonstrate and analyze the performance of mean-field and beyond-mean-field dynamics techniques for the H₂ molecule in one dimension, in the later case capturing the vibronic structure quite accurately, including quantum Franck-Condon effects. In a practical application of this methodology we simulate the absorption spectrum of benzene in full dimensionality using time-dependent density functional theory at the multitrajectory Ehrenfest level, finding good qualitative agreement with experiment and significant spectral reweighting compared to commonly used single-trajectory Ehrenfest dynamics. These results form the foundation for nonlinear spectral calculations and show promise for future application in capturing phenomena associated with vibronic coupling in more complex molecular and potentially condensed phase systems.

The quantum-Ehrenfest method with the inclusion of an IR pulse: Application to electron dynamics of the allene radical cation

We describe the implementation of a laser control pulse in the quantum-Ehrenfest method, a molecular quantum dynamics method that solves the time-dependent Schrödinger equation for both electrons and nuclei. The oscillating electric field-dipole interaction is incorporated directly in the one-electron Hamiltonian of the electronic structure part of the algorithm. We then use the coupled electron-nuclear dynamics of the π-system in the allene radical cation (•CH₂=C=CH₂)⁺ as a simple model of a pump-control experiment. We start (pump) with a two-state superposition of two cationic states. The resulting electron dynamics corresponds to the rapid oscillation of the unpaired electron between the two terminal methylenes. This electron dynamics is, in turn, coupled to the torsional motion of the terminal methylenes. There is a conical intersection at 90° twist, where the electron dynamics collapses because the adiabatic states become degenerate. After passing the conical intersection, the electron dynamics revives. The IR pulse (control) in our simulations is timed to have its maximum at the conical intersection. Our simulations show that the effect of the (control) pulse is to change the electron dynamics at the conical intersection and, as a consequence, the concomitant nuclear dynamics, which is dominated by the change in the torsional angle.

Stimulated X-ray Resonant Raman Spectroscopy of Conical Intersections in Thiophenol

The conical intersection dynamics of thiophenol is studied by computing the stimulated X-ray resonant Raman spectroscopy signals. The hybrid probing field is constructed of a hard X-ray narrowband femtosecond pulse combined with an attosecond broadband X-ray pulse to provide optimal spectral and temporal resolutions for electronic coherences in the level crossing region. The signal carries phase information about the valence-core electronic coupling in the vicinity of conical intersections. Two conical intersections occurring during the course of the S-H dissociation dynamics can be distinguished by their valence-core transition frequencies computed at the complete active space self-consistent field level. The X-ray pulse is tuned such that the Raman transition at the first conical intersection between ¹πσ* and 1¹ππ* involves higher core levels, while the Raman transition at the second conical intersection between ¹πσ* and S₀ involves the lowest core level in the sulfur K-edge.

Stimulated X-ray Raman Imaging of Conical Intersections

The conical intersection dynamics of thiophenol is studied theoretically using the stimulated X-ray Raman imaging (SXRI) technique. SXRI employs a hard X-ray narrowband/broadband hybrid probe field and provides a real-time and real-space image of the passage through conical intersections. The signal, calculated using the minimal-coupling radiation/matter Hamiltonian, carries the phase information, and the real-space image of the transition charge density can be reconstructed by its Fourier transform. The two conical intersections (S₂/S₁ (1¹ππ*/¹πσ*) and S₁/S₀ (¹πσ*/S₀)) can be distinguished and identified by the diffraction patterns in the level crossing regimes.

Effects of probe energy and competing pathways on time-resolved photoelectron spectroscopy: the ring-opening of 1,3-cyclohexadiene

Photoelectron spectra for the ring-opening dynamics of 1,3-cyclohexadiene are studied using a model based on quantum molecular dynamics and the Dyson orbital approach.

On the importance of initial conditions for excited-state dynamics

The vast majority of ab initio excited-state simulations are performed within semiclassical, trajectory-based approaches. Apart from the underlying electronic-structure theory, the reliability of the simulations is controlled by a selection of initial conditions for the classical trajectories. We discuss appropriate choices of initial conditions for simulations of different experimental arrangements: dynamics initiated by continuum-wave (CW) laser fields or triggered by ultrashort laser pulses.

Ab initio calculation of inelastic scattering

We calculate nonresonant inelastic electron and X-ray scattering cross sections for bound-to-bound transitions in atoms and molecules from ab initio electronic wavefunctions.

Toward fully quantum modelling of ultrafast photodissociation imaging experiments. Treating tunnelling in the ab initio multiple cloning approach

We present an account of our recent effort to improve simulation of the photodissociation of small heteroaromatic molecules using the Ab Initio Multiple Cloning (AIMC) algorithm. The ultimate goal is to create a quantitative and converged technique for fully quantum simulations which treats both electrons and nuclei on a fully quantum level. We calculate and analyse the total kinetic energy release (TKER) spectra and Velocity Map Images (VMI), and compare the results directly with experimental measurements. In this work, we perform new extensive calculations using an improved AIMC algorithm that now takes into account the tunnelling of hydrogen atoms. This can play an extremely important role in photodissociation dynamics.

Non-adiabatic excited state molecular dynamics of phenylene ethynylene dendrimer using a multiconfigurational Ehrenfest approach

Photoinduced dynamics of electronic and vibrational unidirectional energy transfer between meta-linked building blocks in a phenylene ethynylene dendrimer is simulated using a multiconfigurational Ehrenfest in time-dependent diabatic basis (MCE-TDDB) method.

Imaging Molecular Motion: Femtosecond X-Ray Scattering of an Electrocyclic Chemical Reaction

Structural rearrangements within single molecules occur on ultrafast time scales. Many aspects of molecular dynamics, such as the energy flow through excited states, have been studied using spectroscopic techniques, yet the goal to watch molecules evolve their geometrical structure in real time remains challenging. By mapping nuclear motions using femtosecond x-ray pulses, we have created real-space representations of the evolving dynamics during a well-known chemical reaction and show a series of time-sorted structural snapshots produced by ultrafast time-resolved hard x-ray scattering. A computational analysis optimally matches the series of scattering patterns produced by the x rays to a multitude of potential reaction paths. In so doing, we have made a critical step toward the goal of viewing chemical reactions on femtosecond time scales, opening a new direction in studies of ultrafast chemical reactions in the gas phase.

Ab initio multiple cloning simulations of pyrrole photodissociation: TKER spectra and velocity map imaging

We report a detailed computational simulation of the photodissociation of pyrrole using the ab initio Multiple Cloning (AIMC) method implemented within MOLPRO.