Isomerization Through Conical Intersections

Quantum Computation of Conical Intersections on a Programmable Superconducting Quantum Processor

Conical intersections (CIs) are pivotal in many photochemical processes. Traditional quantum chemistry methods, such as the state-average multiconfigurational methods, face computational hurdles in solving the electronic Schrödinger equation within the active space on classical computers. While quantum computing offers a potential solution, its feasibility in studying CIs, particularly on real quantum hardware, remains largely unexplored. Here, we present the first successful realization of a hybrid quantum-classical state-average complete active space self-consistent field method based on the variational quantum eigensolver (VQE-SA-CASSCF) on a superconducting quantum processor. This approach is applied to investigate CIs in two prototypical systems─ethylene (C2H4) and triatomic hydrogen (H3). We illustrate that VQE-SA-CASSCF, coupled with ongoing hardware and algorithmic enhancements, can lead to a correct description of CIs on existing quantum devices. These results lay the groundwork for exploring the potential of quantum computing to study CIs in more complex systems in the future.

The development of the QM/MM interface and its application for the on-the-fly QM/MM nonadiabatic dynamics in JADE package: Theory, implementation, and applications

Understanding the nonadiabatic dynamics of complex systems is a challenging task in computational photochemistry. Herein, we present an efficient and user-friendly quantum mechanics/molecular mechanics (QM/MM) interface to run on-the-fly nonadiabatic dynamics. Currently, this interface consists of an independent set of codes designed for general-purpose use. Herein, we demonstrate the ability and feasibility of the QM/MM interface by integrating it with our long-term developed JADE package. Tailored to handle nonadiabatic processes in various complex systems, especially condensed phases and protein environments, we delve into the theories, implementations, and applications of on-the-fly QM/MM nonadiabatic dynamics. The QM/MM approach is established within the framework of the additive QM/MM scheme, employing electrostatic embedding, link-atom inclusion, and charge-redistribution schemes to treat the QM/MM boundary. Trajectory surface-hopping dynamics are facilitated using the fewest switches algorithm, encompassing classical and quantum treatments for nuclear and electronic motions, respectively. Finally, we report simulations of nonadiabatic dynamics for two typical systems: azomethane in water and the retinal chromophore PSB3 in a protein environment. Our results not only illustrate the power of the QM/MM program but also reveal the important roles of environmental factors in nonadiabatic processes.

Development of Parallel On-the-Fly Algorithm for Global Exploration of Conical Intersection Seam Space

Conical intersection (CI) seams are configuration spaces of a molecular system where two or more (spin) adiabatic electronic states are degenerate in energy. They play essential roles in photochemistry because nonradiative decays often occur near the minima of the seam, i.e., the minimum energy CIs (MECIs). Thus, it is important to explore the CI seams and discover the MECIs. Although various approaches exist for CI seam exploration, most of them are local in nature, requiring reasonable initial guesses of geometries and nuclear gradients during the search. Global search algorithms, on the other hand, are powerful because they can fully sample the configurational space and locate important MECIs missed by local algorithms. However, global algorithms are often computationally expensive for large systems due to their poor scalability with respect to the number of degrees of freedom. To overcome this challenge, we develop the parallel on-the-fly Crystal algorithm to globally explore the CI seam space, taking advantage of its superior scaling behavior. Specifically, Crystal is coupled with on-the-fly evaluations of the excited and ground state energies using multireference electronic structure methods. Meanwhile, the algorithm is parallelized to further boost its computational efficiency. The effectiveness of this new algorithm is tested for three types of molecular photoswitches of significant importance in material and biomedical sciences: photostatin (PST), stilbene, and butadiene. A rudimentary implementation of the algorithm is applied to PST and stilbene, resulting in the discovery of all previously identified MECIs and several new ones. A refined version of the algorithm, combined with a systematic clustering technique, is applied to butadiene, resulting in the identification of an unprecedented number of energetically accessible MECIs. The results demonstrate that the parallel on-the-fly Crystal algorithm is a powerful tool for automated global CI seam exploration.

Generalized Semiclassical Ehrenfest Method: A Route to Wave Function-Free Photochemistry and Nonadiabatic Dynamics with Only Potential Energies and Gradients

We reconsider recent methods by which direct dynamics calculations of electronically nonadiabatic processes can be carried out while requiring only adiabatic potential energies and their gradients. We show that these methods can be understood in terms of a new generalization of the well-known semiclassical Ehrenfest method. This is convenient because it eliminates the need to evaluate electronic wave functions and their matrix elements along the mixed quantum-classical trajectories. The new approximations and procedures enabling this advance are the curvature-driven approximation to the time-derivative coupling, the generalized semiclassical Ehrenfest method, and a new gradient correction scheme called the time-derivative matrix (TDM) scheme. When spin-orbit coupling is present, one can carry out dynamics calculations in the fully adiabatic basis using potential energies and gradients calculated without spin-orbit coupling plus the spin-orbit coupling matrix elements. Even when spin-orbit coupling is neglected, the method is useful because it allows calculations by electronic structure methods for which nonadiabatic coupling vectors are unavailable. In order to place the new considerations in context, the article starts out with a review of background material on trajectory surface hopping, the semiclassical Ehrenfest scheme, and methods for incorporating decoherence. We consider both internal conversion and intersystem crossing. We also review several examples from our group of successful applications of the curvature-driven approximation.

Regioselective and solvent-dependent photoisomerization induced internal conversion in red fluorescent protein chromophore analogues

Photophysical properties of three red fluorescent protein (RFP) chromophore analogues are reported here. The three RFP chromophore analogues differ in the additional conjugation present in the RFP chromophore. The three chromophores do not exhibit any solvent effect in both absorption and fluorescence spectra. The photoirradiation experiments and recording of 1H NMR before and after irradiation on one of the three RFP model chromophores show isomerization of the (Z,E) diastereomer to the (E,E) diastereomer. Calculation of S0 and S1 potential energy curves shows the preference for isomerization through the exocyclic C[double bond, length as m-dash]C bond with Z-stereochemistry, thus corroborating the experimental results. The computational studies also suggest that torsional motion along the exocyclic C[double bond, length as m-dash]C bond pushes the molecules to a conical intersection, thus paving the pathway for radiationless deactivation.

Quantum Quality with Classical Cost: Ab Initio Nonadiabatic Dynamics Simulations Using the Mapping Approach to Surface Hopping

Nonadiabatic dynamics methods are an essential tool for investigating photochemical processes. In the context of employing first-principles electronic structure techniques, such simulations can be carried out in a practical manner using semiclassical trajectory-based methods or wave packet approaches. While all approaches applicable to first-principles simulations are necessarily approximate, it is commonly thought that wave packet approaches offer inherent advantages over their semiclassical counterparts in terms of accuracy and that this trait simply comes at a higher computational cost. Here we demonstrate that the mapping approach to surface hopping (MASH), a recently introduced trajectory-based nonadiabatic dynamics method, can be efficiently applied in tandem with ab initio electronic structure. Our results even suggest that MASH may provide more accurate results than on-the-fly wave packet techniques, all at a much lower computational cost.

Excited State Dynamics in Unidirectional Photochemical Molecular Motors

Unidirectional photochemically driven molecular motors (PMMs) convert the energy of absorbed light into continuous rotational motion. As such they are key components in the design of molecular machines. The prototypical and most widely employed class of PMMs is the overcrowded alkenes, where rotational motion is driven by successive photoisomerization and thermal helix inversion steps. The efficiency of such PMMs depends upon the speed of rotation, determined by the rate of ground state thermal helix inversion, and the quantum yield of photoisomerization, which is dependent on the excited state energy landscape. The former has been optimized by synthetic modification across three generations of overcrowded alkene PMMs. These improvements have often been at the expense of photoisomerization yield, where there remains room for improvement. In this perspective we review the application of ultrafast spectroscopy to characterize the excited state dynamics in PMMs. These measurements lead to a general mechanism for all generations of PMMs, involving subpicosecond decay of a Franck-Condon excited state to populate a dark excited state which decays within picoseconds via conical intersections with the electronic ground state. The model is discussed in the context of excited state dynamics calculations. Studies of PMM photochemical dynamics as a function of solvent suggest exploitation of intramolecular charge transfer and solvent polarity as a route to controlling photoisomerization yield. A test of these ideas for a first generation motor reveals a high degree of solvent control over isomerization yield. These results suggest a pathway to fine control over the performance of future PMMs.

Intramolecular charge transfer and the function of vibronic excitons in photosynthetic light harvesting

A widely discussed explanation for the prevalence of pairs or clusters of closely spaced electronic chromophores in photosynthetic light-harvesting proteins is the presence of ultrafast and highly directional excitation energy transfer pathways mediated by vibronic excitons, the delocalized optical excitations derived from mixing of the electronic and vibrational states of the chromophores. We discuss herein the hypothesis that internal conversion processes between exciton states on the <100 fs timescale are possible when the excitonic potential energy surfaces are controlled by the vibrational modes that induce charge transfer character in a strongly coupled system of chromophores. We discuss two examples, the peridinin-chlorophyll protein from marine dinoflagellates and the intact phycobilisome from cyanobacteria, in which the intramolecular charge-transfer (ICT) character arising from out-of-plane distortion of the conjugation of carotenoid or bilin chromophores also results in localization of the initially delocalized optical excitation on the vibrational timescale. Tuning of the ground state conformations of the chromophores to manipulate their ICT character provides a natural photoregulatory mechanism, which would control the overall quantum yield of excitation energy transfer by turning on and off the delocalized character of the optical excitations.

Quantum simulation of conical intersections

We explore the simulation of conical intersections (CIs) on quantum devices, setting the groundwork for potential applications in nonadiabatic quantum dynamics within molecular systems. The intersecting potential energy surfaces of H3+ are computed from a variance-based contracted quantum eigensolver. We show how the CIs can be correctly described on quantum devices using wavefunctions generated by the anti-Hermitian contracted Schrödinger equation ansatz, which is a unitary transformation of wavefunctions that preserves the topography of CIs. A hybrid quantum-classical procedure is used to locate the seam of CIs. Additionally, we discuss the quantum implementation of the adiabatic to diabatic transformation and its relation to the geometric phase effect. Results on noisy intermediate-scale quantum devices showcase the potential of quantum computers in dealing with problems in nonadiabatic chemistry.

Can classical mechanics sense conical intersection?

Conical intersection (CI) leads to fast electronic energy transfer. However, Hamm and Stock [Phys. Rev. Lett. 109, 173201 (2012)] showed the existence of a vibrational CI and its role in vibrational energy relaxation. In this paper, we further investigate the vibrational energy relaxation using an isolated model Hamiltonian system of four vibrational modes with two distinctively different timescales (two fast modes and two slow modes). We show that the excitation of the slow modes plays a crucial role in the energy relaxation mechanism. We also analyze the system from a mixed quantum-classical (surface hopping method) and a completely classical point of view. Notably, surface hopping and even classical simulations also capture fast energy relaxation, which is a signature of CI's existence.

Numerical Equivalence of Diabatic and Adiabatic Representations in Diatomic Molecules

The (time-independent) Schrödinger equation for atomistic systems is solved by using the adiabatic potential energy curves (PECs) and the associated adiabatic approximation. In cases where interactions between electronic states become important, the associated nonadiabatic effects are taken into account via derivative couplings (DDRs), also known as nonadiabatic couplings (NACs). For diatomic molecules, the corresponding PECs in the adiabatic representation are characterized by avoided crossings. The alternative to the adiabatic approach is the diabatic representation obtained via a unitary transformation of the adiabatic states by minimizing the DDRs. For diatomics, the diabatic representation has zero DDR and nondiagonal diabatic couplings ensue. The two representations are fully equivalent and so should be the rovibronic energies and wave functions, which result from the solution of the corresponding Schrödinger equations. We demonstrate (for the first time) the numerical equivalence between the adiabatic and diabatic rovibronic calculations of diatomic molecules using the ab initio curves of yttrium oxide (YO) and carbon monohydride (CH) as examples of two-state systems, where YO is characterized by a strong NAC, while CH has a strong diabatic coupling. Rovibronic energies and wave functions are computed using a new diabatic module implemented in the variational rovibronic code Duo. We show that it is important to include both the diagonal Born-Oppenheimer correction and nondiagonal DDRs. We also show that the convergence of the vibronic energy calculations can strongly depend on the representation of nuclear motion used and that no one representation is best in all cases.

Probing Conical Intersection in the Multipathway Isomerization of CH3Cl Using Coulomb Explosion

Ubiquitous ultrafast isomerization is paramount in photoexcited molecules, in which non-adiabatic coupling among multiple electronic states can occur. We use the pump-probe Coulomb explosion imaging method to study the isomerization of CH3Cl molecules. We find that the isomerization under our strong field pump-probe scheme proceeds along multiple pathways, which are encoded in several distinct branches of the time-resolved kinetic energy release spectra for the CH2++HCl+ Coulomb explosion channel. Apart from the isomerized dissociative pathway in neutral and cationic excited states, the pump laser can also induce coherent vibrational dynamics in two coupled intermediate states and set up the initial conditions for the two concurrently proceeding isomerization pathways. The isomerization of CH3Cl provides an intriguing example of a chemical reaction consisting of multiple pathways and non-adiabatic dynamics.

Signatures of Conical Intersections in Extreme Ultraviolet Photoelectron Spectra of Furan Measured with 15 fs Time Resolution

Ultrafast internal conversion of furan upon deep UV excitation at 200 nm is studied by using extreme ultraviolet time-resolved photoelectron spectroscopy with a time resolution of 15 fs. Ballistic nuclear wavepacket motion from the 1B2(ππ*) state to the ground state is fully observed using 21.7 eV probe pulses. Through the performance of a comparison with the results of electronic structure calculations at the MS(3)-CASPT2(10,10)/cc-pVTZ level of theory, the photoelectron signals from the conical intersection regions are identified.

Raman Spectroscopy of Conical Intersections Using Entangled Photons

Ultrafast Raman spectroscopy with attosecond pulses in the extreme ultraviolet and X-ray regime has been proposed theoretically for tracking the non-adiabatic dynamics of molecules in great detail. The large bandwidth of these pulses, which span several electronvolts within a couple of femtoseconds, provides a unique tool for tracking non-adiabatic phenomena. However, spectroscopy with classical light is limited by the time-bandwidth product of the probe laser pulse. In this work, we theoretically investigate an ultrafast Raman spectroscopy scheme that utilizes pairs of entangled photons. Our model simulations demonstrate that the dynamics in the vicinity of a conical intersection can be resolved with unprecedented resolution in the time and frequency domain.

Solvent-polarity dependence of ultrafast excited-state dynamics of trans-4-nitrostilbene

Ultrafast excited-state dynamics of the simplest nitrostilbenes, namely trans-4-nitrostilbene (t-NSB), was studied in solvents of various polarities with ultrafast broadband time-resolved fluorescence and transient absorption spectroscopies, and by quantum-chemical computations. The results revealed that the initially excited S1(ππ*) state deactivation dynamics is strongly influenced by the solvent polarity. Specifically, the t-NSB S1-state lifetime decreases by three orders of magnitude from ∼60 ps in high-polarity solvents to ∼60 fs in nonpolar solvents. The strong solvent-polarity dependence arises from the differences in dipole moments among the S1 and relevant states, including the major intersystem crossing (ISC) receiver triplet states, and therefore, the solvent polarity can modulate their relative energies and ISC rates. In nonpolar solvents, the sub-100 fs lifetime is due to a combination of efficient ISC and internal conversion. In medium-polarity solvents, the S1-state population decays via a competing ISC relaxation mechanism in a biphasic manner, and the ISC rates are found to obey the inverse energy gap law of the strong coupling case. In high-polarity solvents, the S1 state is stabilized to a much lower energy such that ISC becomes energetically infeasible, and the S1 state decays via barrier crossing along the torsion angle of the central ethylenic bond to the nonfluorescent perpendicular configuration. Regardless of the initial S1-state deactivation pathways in various solvents, the excited-state population is ultimately trapped in the metastable T1-state perpendicular configuration, at which a slower ISC occurs to bring the system to the ground state and bifurcate into either trans or cis form of NSB.

Photoelectron spectra of benzene: Can path dependent diabatic surfaces provide unique observables?

While carrying out Beyond Born-Oppenheimer theory based diabatization, the solutions of adiabatic-to-diabatic transformation equations depend on the paths of integration over two-dimensional cross-sections of multi-dimensional space of nuclear degrees of freedom. It is shown that such path-dependent solutions leading to diabatic potential energy surface matrices computed along any two different paths are related through an orthogonal matrix, and thereby, those surface matrices should provide unique observables. While exploring the numerical validity of the theoretical framework, we construct diabatic Hamiltonians for the five low-lying electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of benzene radical cation (C6H6+) along three different approaches of contour integration over two dimensional nuclear planes constituted by seven non-adiabatically active normal modes. Three different diabatic surface matrices are further employed to generate the photoelectron spectra of the benzene molecule (C6H6). It is interesting to note that the spectral peak positions and intensity patterns for all three cases are almost close to each other and also exhibit very good agreement with the experimental results.

On the description of conical intersections between excited electronic states with LR-TDDFT and ADC(2)

Conical intersections constitute the conceptual bedrock of our working understanding of ultrafast, nonadiabatic processes within photochemistry (and photophysics). Accurate calculation of potential energy surfaces within the vicinity of conical intersections, however, still poses a serious challenge to many popular electronic structure methods. Multiple works have reported on the deficiency of methods like linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) or algebraic diagrammatic construction to second-order [ADC(2)]-approaches often used in excited-state molecular dynamics simulations-to describe conical intersections between the ground and excited electronic states. In the present study, we focus our attention on conical intersections between excited electronic states and probe the ability of AA LR-TDDFT and ADC(2) to describe their topology and topography, using protonated formaldimine and pyrazine as two exemplar molecules. We also take the opportunity to revisit the performance of these methods in describing conical intersections involving the ground electronic state in protonated formaldimine-highlighting in particular how the intersection ring exhibited by AA LR-TDDFT can be perceived either as a (near-to-linear) seam of intersection or two interpenetrating cones, depending on the magnitude of molecular distortions within the branching space.

What Controls the Quality of Photodynamical Simulations? Electronic Structure Versus Nonadiabatic Algorithm

The field of nonadiabatic dynamics has matured over the last decade with a range of algorithms and electronic structure methods available at the moment. While the community currently focuses more on developing and benchmarking new nonadiabatic dynamics algorithms, the underlying electronic structure controls the outcome of nonadiabatic simulations. Yet, the electronic-structure sensitivity analysis is typically neglected. In this work, we present a sensitivity analysis of the nonadiabatic dynamics of cyclopropanone to electronic structure methods and nonadiabatic dynamics algorithms. In particular, we compare wave function-based CASSCF, FOMO-CASCI, MS- and XMS-CASPT2, density-functional REKS, and semiempirical MRCI-OM3 electronic structure methods with the Landau-Zener surface hopping, fewest switches surface hopping, and ab initio multiple spawning with informed stochastic selection algorithms. The results clearly demonstrate that the electronic structure choice significantly influences the accuracy of nonadiabatic dynamics for cyclopropanone even when the potential energy surfaces exhibit qualitative and quantitative similarities. Thus, selecting the electronic structure solely on the basis of the mapping of potential energy surfaces can be misleading. Conversely, we observe no discernible differences in the performance of the nonadiabatic dynamics algorithms across the various methods. Based on the above results, we discuss the present-day practice in computational photodynamics.

Fantastical excited state optimized structures and where to find them

The quantum chemistry community has developed analytic forces for approximate electronic excited states to enable walking on excited state potential energy surfaces (PES). One can thereby computationally characterize excited state minima and saddle points. Always implicit in using this machinery is the fact that an excited state PES only exists within the realm of the Born-Oppenheimer approximation, where the nuclear and electronic degrees of freedom separate. This work demonstrates through ab initio calculations and simple nonadiabatic dynamics that some excited state minimum structures are fantastical: they appear to exist as stable configurations only as a consequence of the PES construct, rather than being physically observable. Each fantastical structure exhibits an unphysically high predicted harmonic frequency and associated force constant. This fact can serve as a valuable diagnostic of when an optimized excited state structure is non-observable. The origin of this phenomenon can be attributed to the coupling between different electronic states. As PESs approach one another, the upper surface can form a minimum that is very close to a near-touching point. The force constant, evaluated at this minimum, relates to the strength of the electronic coupling rather than to any characteristic excited state vibration. Nonadiabatic dynamics results using a Landau-Zener model illustrate that fantastical excited state structures have extremely short lifetimes on the order of a few femtoseconds. Their appearance in a calculation signals the presence of a nearby conical intersection through which the system will rapidly cross to a lower surface.

An energy decomposition and extrapolation scheme for evaluating electron transfer rate constants: a case study on electron self-exchange reactions of transition metal complexes

A simple approach to the analysis of electron transfer (ET) reactions based on energy decomposition and extrapolation schemes is proposed. The present energy decomposition and extrapolation-based electron localization (EDEEL) method represents the diabatic energies for the initial and final states using the adiabatic energies of the donor and acceptor species and their complex. A scheme for the efficient estimation of ET rate constants is also proposed. EDEEL is semi-quantitative by directly evaluating the seam-of-crossing region of two diabatic potentials. In a numerical test, EDEEL successfully provided ET rate constants for electron self-exchange reactions of thirteen transition metal complexes with reasonable accuracy. In addition, its energy decomposition and extrapolation schemes provide all the energy values required for activation-strain model (ASM) analysis. The ASM analysis using EDEEL provided rational interpretations of the variation of the ET rate constants as a function of the transition metal complexes. These results suggest that EDEEL is useful for efficiently evaluating ET rate constants and obtaining a rational understanding of their magnitudes.

On the Fluorescence Properties and Nonradiative Transitions in Medium-Sized All-Trans Polyenes

The nonadiabatic photodynamics of all-trans linear polyenes with N = 4-8 conjugated double bonds is studied from an electronic structure perspective. Excitation energies and stationary points for the 1Bu and 2Ag singlet states have been computed by using the state-average complete active space (SA-CASSCF) method and its second-order perturbation theory variant (MS-CASPT2). The dependence of the two low-lying excited states on the "chain length" N has been elucidated. In addition, the 1Bu-2Ag crossing seam has been mapped out in a suitable two-dimensional coordinate space and its minimum within the subspace has been determined. This minimum is found to increase substantially and monotonously in energy with increasing N. This increase is discussed and interpreted in relation to the fluorescence properties of these systems. In particular, it allows to understand the crossover from S1(2Ag) fluorescence for smaller N to S2(1Bu) (or dual) fluorescence for larger N.

Optical Control of Crossing the Conical Intersection in β-Carotene

Optical control of dynamic processes has been challenging yet has only been demonstrated in several chemical and biological systems. The control of a reaction passing the widely present conical intersection has not been realized. Here, we modulated the phase of the excitation pulse to control the dynamics of β-carotene through accessing the conical intersection (CI). We observed different dynamics in 110-220 fs into the CI and the consecutive process in 400-600 fs through another CI by various chirped excitation pulses. We successfully controlled those ultrafast wavepacket dynamics passing the CIs on the femtosecond time scales. The method developed here can be used to control a various of ultrafast chemical and biological reactions through the CI(s).

Competing quantum effects in heavy-atom tunnelling through conical intersections

Thermally activated chemical reactions are typically understood in terms of overcoming potential-energy barriers. However, standard rate theories break down in the presence of a conical intersection (CI) because these processes are inherently nonadiabatic, invalidating the Born-Oppenheimer approximation. Moreover, CIs give rise to intricate nuclear quantum effects such as tunnelling and the geometric phase, which are neglected by standard trajectory-based simulations and remain largely unexplored in complex molecular systems. We present new semiclassical transition-state theories based on an extension of golden-rule instanton theory to describe nonadiabatic tunnelling through CIs and thus provide an intuitive picture for the reaction mechanism. We apply the method in conjunction with first-principles electronic-structure calculations to the electron transfer in the bis(methylene)-adamantyl cation. Our study reveals a strong competition between heavy-atom tunnelling and geometric-phase effects.

Nonadiabatic simulations of photoisomerization and dissociation in ethylene using ab initio classical trajectories

We simulate the nonadiabatic dynamics of photo-induced isomerization and dissociation in ethylene using ab initio classical trajectories in an extended phase space of nuclear and electronic variables. This is achieved by employing the linearized semiclassical initial value representation method for nonadiabatic dynamics, where discrete electronic states are mapped to continuous classical variables using either the Meyer-Miller-Stock-Thoss representation or a more recently introduced spin mapping approach. Trajectory initial conditions are sampled by constraining electronic state variables to a single initial excited state and by drawing nuclear phase space configurations from a Wigner distribution at a finite temperature. An ensemble of classical ab initio trajectories is then generated to compute thermal population correlation functions and analyze the mechanisms of isomerization and dissociation. Our results serve as a demonstration that this parameter-free semiclassical approach is computationally efficient and accurate, identifying mechanistic pathways in agreement with previous theoretical studies and also uncovering dissociation pathways observed experimentally.

Non-adiabatic coupling in the potential energy surfaces of SO2 molecule

To investigate the potential energy surfaces and the coupling between the adiabatic states of SO2 molecules, it is necessary to consider the non-adiabatic coupling terms (NACTs), where the Born-Oppenheimer approximation breaks down. In this work, we analyze the conical intersections between 1 1A1 and 1 1B2 states (the A' states in Cs symmetry) and 1 1A2 and 1 1B1 states (the A'' states in Cs symmetry) using NACTs and adiabatic-to-diabatic transformation (ADT) angles. Our results confirm reasonable interaction between 1 1A1 and 1 1B2 states and strong interaction between 1 1A2 and 1 1B1 states.

Zeno and Anti-Zeno Effects in Nonadiabatic Molecular Dynamics

Decoherence plays an important role in nonadiabatic (NA) molecular dynamics (MD) simulations because it provides a physical mechanism for trajectory hopping and can alter transition rates by orders of magnitude. Generally, decoherence effects slow quantum transitions, as exemplified by the quantum Zeno effect: in the limit of infinitely fast decoherence, the transitions stop. If the measurements are not sufficiently frequent, an opposite quantum anti-Zeno effect occurs, in which the transitions are accelerated with faster decoherence. Using two common NA-MD approaches, fewest switches surface hopping and decoherence-induced surface hopping, combined with analytic examination, we demonstrate that including decoherence into NA-MD slows down NA transitions; however, many realistic systems operate in the anti-Zeno regime. Therefore, it is important that NA-MD methods describe both Zeno and anti-Zeno effects. Numerical simulations of charge trapping and relaxation in graphitic carbon nitride suggest that time-dependent NA Hamiltonians encountered in realistic systems produce robust results with respect to errors in the decoherence time, a favorable feature for NA-MD simulations.

Deciphering Methylation Effects on S2(ππ*) Internal Conversion in the Simplest Linear α,β-Unsaturated Carbonyl

Chemical substituents can influence photodynamics by altering the location of critical points and the topography of the potential energy surfaces (electronic effect) and by selectively modifying the inertia of specific nuclear modes (inertial effects). Using nonadiabatic dynamics simulations, we investigate the impact of methylation on S₂(ππ*) internal conversion in acrolein, the simplest linear α,β-unsaturated carbonyl. Consistent with time constants reported in a previous time-resolved photoelectron spectroscopy study, S₂ → S₁ deactivation occurs on an ultrafast time scale (∼50 fs). However, our simulations do not corroborate the sequential decay model used to fit the experiment. Instead, upon reaching the S₁ state, the wavepacket bifurcates: a portion undergoes ballistic S₁ → S₀ deactivation (∼90 fs) mediated by fast bond-length alternation motion, while the remaining decays on the picosecond time scale. Our analysis reveals that methyl substitution, generally assumed to mainly exert inertial influence, is also manifested in important electronic effects due to its weak electron-donating ability. While methylation at the β C atom gives rise to effects principally of an inertial nature, such as retarding the twisting motion of the terminal -CHCH₃ group and increasing its coupling with pyramidalization, methylation at the α or carbonyl C atom modifies the potential energy surfaces in a way that also contributes to altering the late S₁-decay behavior. Specifically, our results suggest that the observed slowing of the picosecond component upon α-methylation is a consequence of a tighter surface and reduced amplitude along the central pyramidalization, effectively restricting the access to the S₁/S₀-intersection seam. Our work offers new insight into the S₂(ππ*) internal conversion mechanisms in acrolein and its methylated derivatives and highlights site-selective methylation as a tuning knob to manipulate photochemical reactions.

Scoring molecular wires subject to an ultrafast laser pulse for molecular electronic devices

A nonionizing ultrafast laser pulse of 20-fs duration with a peak amplitude electric-field ±E = 200 × 10^-4 a.u. was simulated. It was applied to the ethene molecule to consider its effect on the electron dynamics, both during the application of the laser pulse and for up to 100 fs after the pulse was switched off. Four laser pulse frequencies ω = 0.2692, 0.2808, 0.2830, and 0.2900 a.u. were chosen to correspond to excitation energies mid-way between the (S₁ ,S₂ ), (S₂ ,S₃ ), (S₃ ,S₄ ) and (S₄ ,S₅ ) electronic states, respectively. Scalar quantum theory of atoms in molecules (QTAIM) was used to quantify the shifts of the C1C2 bond critical points (BCPs). Depending on the frequencies ω selected, the C1C2 BCP shifts were up to 5.8 times higher after the pulse was switched off compared with a static E-field with the same magnitude. Next generation QTAIM (NG-QTAIM) was used to visualize and quantify the directional chemical character. In particular, polarization effects and bond strengths, in the form of bond-rigidity vs. bond-flexibility, were found, for some laser pulse frequencies, to increase after the laser pulse was switched off. Our analysis demonstrates that NG-QTAIM, in partnership with ultrafast laser irradiation, is useful as a tool in the emerging field of ultrafast electron dynamics, which will be essential for the design, and control of molecular electronic devices.

New Gradient Correction Scheme for Electronically Nonadiabatic Dynamics Involving Multiple Spin States

It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.

Ultrafast Imaging of the Jahn-Teller Topography in Carbon Tetrachloride

We report the direct time-domain observation of ultrafast dynamics driven by the Jahn-Teller effect. Using time-resolved photoelectron spectroscopy with a vacuum-ultraviolet femtosecond source to prepare high-lying Rydberg states of carbon tetrachloride, our measurements reveal the local topography of a Jahn-Teller conical intersection. The pump pulse prepares a configurationally mixed superposition of the degenerate ¹T₂ 4p-Rydberg states, which then distorts through spontaneous symmetry breaking that we identify to follow the t₂ bending motion. Photoionization of these states to three cationic states ²T₁, ²T₂, and ²E reveals a shift in the center-of-mass of the photoelectron peaks associated with the ²T_n states which reveals the local topography of the Jahn-Teller conical intersection region prepared by the pump pulse. Time-dependent density functional theory calculations confirm that the dominant nuclear motion observed in the spectrum is the CCl₄ t₂ bending mode. The large density of states in the VUV spectral region at 9.33 eV of carbon tetrachloride and strong vibronic coupling result in ultrafast decay of the excited-state signal with a time constant of 75(4) fs.

A Nitrogen Out-of-Plane (NOOP) Mechanism for Imine-Based Light-Driven Molecular Motors

Light-driven molecular motors have generated considerable interest due to their potential applications in material and biological systems. Recently, Greb and Lehn reported a new class of molecular motors, chiral N-alkyl imines, which undergo unidirectional rotation induced by light and heat. The mechanism of unidirectional motion in molecular motors containing a C═N group has been assumed to consist of photoinduced torsion about the double bond. In this work, we present a computational study of the photoisomerization dynamics of a chiral N-alkyl imine motor. We find that the location and energetics of minimal energy conical intersections (MECIs) alone are insufficient to understand the mechanism of the motor. Furthermore, a key part of the mechanism consists of out-of-plane distortions of the N atom (followed by isomerization about the double bond). Dynamic effects and out-of-plane distortions are critical to understand the observed (rather low) quantum yield for photoisomerization. Our results provide hints as to how the photoisomerization quantum yield might be increased, improving the efficiency of this class of molecular motors.

Revisiting photocyclization of the donor-acceptor stenhouse adduct: missing pieces in the mechanistic jigsaw discovered

Donor-acceptor Stenhouse adducts (DASA) have recently emerged as a class of visible-light-induced photochromic molecular switches, but their photocyclization mechanism remains puzzling and incomplete. In this work, we carried out MS-CASPT2//SA-CASSCF calculations to reveal the complete mechanism of the dominant channels and possible side reactions. We found that a new thermal-then-photo isomerization channel, i.e., EEZ → EZZ → EZE, other than the commonly accepted EEZ → EEE → EZE channel, is dominant in the initial step. Besides, our calculations rationalized why the expected byproducts ZEZ and ZEE are unobserved and proposed a competitive stepwise channel for the final ring-closure step. The findings here redraw the mechanistic picture of the DASA reaction by better accounting for experimental observations, and more importantly, provide critical physical insight in understanding the interplay between thermal- and photo-induced processes widely present in photochemical synthesis and reactions.

Matrix isolation and photorearrangement of cis- and trans-1,2-ethenediol to glycolaldehyde

1,2-Ethenediols are deemed key intermediates in prebiotic and interstellar syntheses of carbohydrates. Here we present the gas-phase synthesis of these enediols, the high-energy tautomers of glycolaldehyde, trapped in cryogenic argon matrices. Importantly, upon photolysis at λ = 180-254 nm, the enols rearrange to the simplest sugar glycolaldehyde.

Ultrafast Ring Closure Reaction of Gaseous cis-Stilbene from S1(ππ*)

cis-Stilbene (cis-St) is a well-known benchmark system for cis-trans photoisomerization. cis-St also produces 4a,4b-dihydrophenanthrene (DHP) in solution with a quantum yield of less than 0.19. The ring closure reaction, however, has never been identified for gaseous cis-St, and a recent computational simulation predicted the quantum yield of DHP to be only 0.04. In the present study, we identified an ultrafast ring closure reaction of gaseous cis-St for the first time using extreme ultraviolet time-resolved photoelectron spectroscopy. Surface hopping trajectory calculations at the SA3-XMS-CASPT2(2,2) level of theory reproduce the features of the observed time-resolved photoelectron spectra and predict the cis-St:DHP:trans-St branching ratio to be 0.55:0.41:0.04, in contrast with previous estimates. The results indicate that photoexcited cis-St favors ring closure over cis-trans isomerization under the isolated condition.

Nonadiabatic Dynamics in a Continuous Circularly Polarized Laser Field with Floquet Phase-Space Surface Hopping

Nonadiabatic chemical reactions involving continuous circularly polarized light (cw CPL) have not attracted as much attention as dynamics in unpolarized/linearly polarized light. However, including circularly (in contrast to linearly) polarized light allows one to effectively introduce a complex-valued time-dependent Hamiltonian, which offers a new path for control or exploration through the introduction of Berry forces. Here, we investigate several inexpensive semiclassical approaches for modeling such nonadiabatic dynamics in the presence of a time-dependent complex-valued Hamiltonian, beginning with a straightforward instantaneous adiabatic fewest-switches surface hopping (IA-FSSH) approach (where the electronic states depend on position and time), continuing to a standard Floquet fewest switches surface hopping (F-FSSH) approach (where the electronic states depend on position and frequency), and ending with an exotic Floquet phase-space surface hopping (F-PSSH) approach (where the electronic states depend on position, frequency, and momentum). Using a set of model systems with time-dependent complex-valued Hamiltonians, we show that the Floquet phase-space adiabats are the optimal choice of basis as far as accounting for Berry phase effects and delivering accuracy. Thus, the F-PSSH algorithm sets the stage for future modeling of nonadiabatic dynamics under strong externally pumped circular polarization.

Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases

We present a comprehensive excited-state benchmark for the state-averaged (SA) driven similarity renormalization group (DSRG) [Li, C.; Evangelista, F. A. J. Chem. Phys. 2018, 148, 124106]. Following the QUEST database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; Loos, P.-F. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2021, 11, e1517], 280 vertical transition energies of 35 medium-sized molecules are computed using the SA-DSRG derived second- and third-order perturbation theories (PT2/PT3) along with a nonperturbative approach [sq-LDSRG(2)]. Comparing to the theoretical best estimates, the optimal flow parameter is found to be 0.35 and 2.0 Eh-2 for SA-DSRG-PT2 and SA-DSRG-PT3, respectively. For SA-sq-LDSRG(2), a flow parameter of 1.5 Eh-2 provides converged equations without compromising the accuracy. We then assess the accuracy of the SA-DSRG hierarchy using these parameters. The SA-DSRG-PT2 scheme outperforms the level-shifted CASPT2 by 0.10 eV in mean absolute error (MAE), yet this accuracy is slightly inferior than that of CASPT2 with the ionization-potential-electron-affinity shift. Both SA-DSRG-PT3 and SA-sq-LDSRG(2) yield a MAE of 0.10 eV, which is comparable to that of CASPT3 (0.09 eV). Finally, we compute vertical excitation energies of several low-lying singlet states of nucleobases. The SA-sq-LDSRG(2) approach provides highly accurate results for π → π* excitations, while n → π* transitions are better described by SA-DSRG-PT3.

Interexcited State Photophysics I: Benchmarking Density Functionals for Computing Nonadiabatic Couplings and Internal Conversion Rate Constants

We present the first benchmarking study of nonadiabatic matrix coupling elements (NACMEs) calculated using different density functionals. Using the S₁ → S₀ transition in perylene solvated in toluene as a case study, we calculate the photophysical properties and corresponding rate constants for a variety of density functionals from each rung of Jacob's ladder. The singlet photoluminescence quantum yield (sPLQY) is taken as a measure of accuracy, measured experimentally here as 0.955. Important quantum chemical parameters such as geometries, absorption, emission, and adiabatic energies, NACMEs, Hessians, and transition dipole moments were calculated for each density functional basis set combination (data set) using density functional theory based multireference configuration interaction (DFT/MRCI) and compared to experiment where possible. We were able to derive simple relations between the TDDFT and DFT/MRCI photophysical properties; with semiempirical damping factors of ∼0.843 ± 0.017 and ∼0.954 ± 0.064 for TDDFT transition dipole moments and energies to DFT/MRCI level approximations, respectively. NACMEs were dominated by out-of-plane derivative components belonging to the center-most ring atoms with weaker contributions from perturbations along the transverse and longitudinal axes. Calculated theoretical spectra compared well to both experiment and literature, with fluorescence lifetimes between 7.1 and 12.5 ns, agreeing within a factor of 2 with experiment. Internal conversion (IC) rates were then calculated and were found to vary wildly between 10⁶-10¹⁶ s^-1 compared with an experimental rate of the order 10⁷ s^-1. Following further testing by mixing data sets, we found a strong dependence on the method used to obtain the Hessian. The 5 characterized data sets ranked in order of most promising are PBE0/def2-TZVP, ωB97XD/def2-TZVP, HCTH407/TZVP, PBE/TZVP, and PBE/def2-TZVP.

From a one-mode to a multi-mode understanding of conical intersection mediated ultrafast organic photochemical reactions

This review discusses how ultrafast organic photochemical reactions are controlled by conical intersections, highlighting that decay to the ground-state at multiple points of the intersection space results in their multi-mode character.

Biological function investigated by time-resolved structure determination

Inspired by recent progress in time-resolved x-ray crystallography and the adoption of time-resolution by cryo-electronmicroscopy, this article enumerates several approaches developed to become bigger/smaller, faster, and better to gain new insight into the molecular mechanisms of life. This is illustrated by examples where chemical and physical stimuli spawn biological responses on various length and time-scales, from fractions of Ångströms to micro-meters and from femtoseconds to hours.

Ultrafast Dynamics of Fatty Acid Photodecarboxylase in Anionic Semiquinone State

Fatty acid photodecarboxylase is a newly identified blue-light driven photoenzyme that catalyzes decarboxylation of fatty acids. The catalytic reaction involves a transient anionic semiquinone of flavin cofactor (FAD^•-) as an intermediate, but photochemical properties of this anionic radical are largely unknown. Here, we have anaerobically produced the wild-type FAP in the FAD^•- state and conducted femtosecond-resolved fluorescence and absorption measurements. We have observed the multiphasic deactivation dynamics of excited states on multiple time scales from a few picoseconds even to a few nanoseconds through conical intersections between various electronic states. Interestingly, the nanosecond components can only be observed from higher electronic excited states. Our results show the complexity of the energy landscapes of various excited states and rule out the occurrence of electron or proton transfer with nearby residue(s) in the active site.

Conical Intersection in Chemiluminescence of Cyclic Peroxides

Chemiluminescence (CL) utilizing chemiexcitation for energy transformation is one of the most highly sensitive and useful analytical techniques. The chemiexcitation is a chemical process of a ground-state reactant producing an excited-state product, in which a nonadiabatic event is facilitated by conical intersections (CIs), the specific molecular geometries where electronic states are degenerated. Cyclic peroxides, especially 1,2-dioxetane/dioxetanone derivatives, are the iconic chemiluminescent substances. In this Perspective, we concentrated on the CIs in the CL of cyclic peroxides. We first present a computational overview on the role of CIs between the ground (S₀) state and the lowest singlet excited (S₁) state in the thermolysis of cyclic peroxides. Subsequently, we discuss the role of the S₀/S₁ CI in the CL efficiency and point out misunderstandings in some theoretical studies on the singlet chemiexcitations of cyclic peroxides. Finally, we address the challenges and future prospects in theoretically calculating S₀/S₁ CIs and simulating the dynamics and chemiexcitation efficiency in the CL of cyclic peroxides.

Characterizing coherences in chemical dynamics with attosecond time-resolved x-ray absorption spectroscopy

Coherence can drive wave-like motion of electrons and nuclei in photoexcited systems, which can yield fast and efficient ways to exert materials’ functionalities beyond the thermodynamic limit. The search for coherent phenomena has been a central topic in chemical physics although their direct characterization is often elusive. Here, we highlight recent advances in time-resolved x-ray absorption spectroscopy (tr-XAS) to investigate coherent phenomena, especially those that utilize the eminent light source of isolated attosecond pulses. The unparalleled time and state sensitivities of tr-XAS in tandem with the unique element specificity render the method suitable to study valence electronic dynamics in a wide variety of materials. The latest studies have demonstrated the capabilities of tr-XAS to characterize coupled electronic–structural coherence in small molecules and coherent light–matter interactions of core-excited excitons in solids. We address current opportunities and challenges in the exploration of coherent phenomena, with potential applications for energy- and bio-related systems, potential crossings, strongly driven solids, and quantum materials. With the ongoing developments in both theory and light sources, tr-XAS holds great promise for revealing the role of coherences in chemical dynamics.

Nonadiabatic transition paths from quantum jump trajectories

We present a means of studying rare reactive pathways in open quantum systems using transition path theory and ensembles of quantum jump trajectories. This approach allows for the elucidation of reactive paths for dissipative, nonadiabatic dynamics when the system is embedded in a Markovian environment. We detail the dominant pathways and rates of thermally activated processes and the relaxation pathways and photoyields following vertical excitation in a minimal model of a conical intersection. We find that the geometry of the conical intersection affects the electronic character of the transition state as defined through a generalization of a committor function for a thermal barrier crossing event. Similarly, the geometry changes the mechanism of relaxation following a vertical excitation. Relaxation in models resulting from small diabatic coupling proceeds through pathways dominated by pure dephasing, while those with large diabatic coupling proceed through pathways limited by dissipation. The perspective introduced here for the nonadiabatic dynamics of open quantum systems generalizes classical notions of reactive paths to fundamentally quantum mechanical processes.

Excitation energy transfer and vibronic coherence in intact phycobilisomes

The phycobilisome is an oligomeric chromoprotein complex that serves as the principal mid-visible light-harvesting system in cyanobacteria. Here we report the observation of excitation-energy-transfer pathways involving delocalized optical excitations of the bilin (linear tetrapyrrole) chromophores in intact phycobilisomes isolated from Fremyella diplosiphon. By using broadband multidimensional electronic spectroscopy with 6.7-fs laser pulses, we are able to follow the progress of excitation energy from the phycoerythrin disks at the ends of the phycobilisome's rods to the C-phycocyanin disks along their length in <600 fs. Oscillation maps show that coherent wavepacket motions prominently involving the hydrogen out-of-plane vibrations of the bilins mediate non-adiabatic relaxation of a manifold of vibronic exciton states. However, the charge-transfer character of the bilins in the allophycocyanin-containing segments localizes the excitations in the core of the phycobilisome, yielding a kinetic bottleneck that enables photoregulatory mechanisms to operate efficiently on the >10-ps timescale.

Nonadiabatic Excited State Dynamics of Organic Chromophores: Take-Home Messages

Nonadiabatic excited state dynamics are important in a variety of processes. Theoretical and experimental developments have allowed for a great progress in this area, while combining the two is often necessary and the best approach to obtain insight into the photophysical behavior of molecules. In this Feature Article we use examples of our recent work combining time-resolved photoelectron spectroscopy with theoretical nonadiabatic dynamics to highlight important lessons we learned. We compare the nonadiabatic excited state dynamics of three different organic molecules with the aim of elucidating connections between structure and dynamics. Calculations and measurements are compared for uracil, 1,3-cyclooctadiene, and 1,3-cyclohexadiene. The comparison highlights the role of rigidity in influencing the dynamics and the difficulty of capturing the dynamics accurately with calculations.

Nuclear Gradient Expressions for Molecular Cavity Quantum ElectrodynamicsSimulations using Mixed Quantum-Classical Methods

We derive a rigorous nuclear gradient for a molecule-cavity hybrid system using the Quantum Electrodynamics Hamiltonian. We treat the electronic-photonic DOFs as the quantum subsystem, and the nuclei as the classical subsystem. Using the adiabatic basis for the electronic DOF and the Fock basis for the photonic DOF, and requiring the total energy conservation of this mixed quantum-classical system, we derived the rigorous nuclear gradient for the molecule-cavity hybrid system, which is naturally connected to the approximate gradient under the Jaynes-Cummings approximation. The nuclear gradient expression can be readily used in any mixed quantum-classical simulations and will allow one to perform the non-adiabatic on-the-fly simulation of polariton quantum dynamics. The theoretical developments in this work could significantly benefit the polariton quantum dynamics community with a rigorous nuclear gradient of the molecule-cavity hybrid system and have a broad impact on the future non-adiabatic simulations of polariton quantum dynamics.

Accurate Non-Adiabatic Couplings from Optimally-Tuned Range-Separated Hybrid Functionals

Precise theoretical calculations of non-adiabatic couplings, which describe the interaction between two Born-Oppenheimer surfaces, are important for the modeling of radiationless decay mechanisms in photochemical processes. Here, we demonstrate that accurate non-adiabatic couplings can be calculated in the framework of linear-response time-dependent density functional theory by using non-empirical, optimally-tuned range-separated hybrid (OT-RSH) functionals. We focus on molecular radicals, in which ultrafast non-radiative decay plays a crucial role, to find that the OT-RSH functional compares well to wave-function based reference data and competes with the accuracy of semi-empirical CAM-B3LYP calculations. Our findings show that the OT-RSH approach provides very accurate non-adiabatic couplings and, therefore, provides a computationally efficient alternative to wave-function based techniques.

Analysis of molecular photomechanical performance using a one-dimensional harmonic model

The photochemical reaction of a molecule leads to a change in the position of its nuclei that can be harnessed to perform mechanical work. Photomechanical materials use this effect to act as light-powered actuators. In this paper, a one-dimensional model based on coupled harmonic potential energy surfaces is developed to describe the photomechanical response of a molecule. This model generates predictions that are qualitatively consistent with standard mechanochemistry models for ground state rate reactions. To analyze the photomechanical process, excited state dynamics like photon absorption and relaxation are included. The model allows us to derive analytical expressions for the work output, blocking force, and absorbed photon-to-work efficiency. The effects of nonadiabatic electronic coupling, unequal frequency potentials, and the cycling efficiency are also analyzed. If the starting state is the stable (lower energy) isomer, it is possible to attain photon-to-work efficiencies up to 55.4%. If initial state is higher in energy, for example a metastable isomer, then one-way efficiencies > 100% are possible due to the release of stored potential energy. Photomechanical materials can be competitive with photovoltaic–piezoelectric combinations in terms of efficiency, but current materials will require substantial improvement before they can approach the theoretical limits.

Graphical abstract