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.

Automatic Clustering of Excited-State Trajectories: Application to Photoexcited Dynamics

We introduce automatic clustering as a computationally efficient tool for classifying and interpreting trajectories from simulations of photo-excited dynamics. Trajectories are treated as time-series data, with the features for clustering selected by variance mapping of normalized data. The L2-norm and dynamic time warping are proposed as suitable similarity measures for calculating the distance matrices, and these are clustered using the unsupervised density-based DBSCAN algorithm. The silhouette coefficient and the number of trajectories classified as noise are used as quality measures for the clustering. The ability of clustering to provide rapid overview of large and complex trajectory data sets, and its utility for extracting chemical and physical insight, is demonstrated on trajectories corresponding to the photochemical ring-opening reaction of 1,3-cyclohexadiene, noting that the clustering can be used to generate reduced dimensionality representations in an unbiased manner.

Real-time observation of the Woodward-Hoffmann rule for 1,3-cyclohexadiene by femtosecond soft X-ray transient absorption

The stereochemistry of pericyclic reactions is explained by orbital symmetry conservation, referred to as the Woodward-Hoffmann (WH) rule. Although this rule has been verified using the structures of reactants and products, the temporal evolution of the orbital symmetry during the reaction has not been clarified. Herein, we used femtosecond soft X-ray transient absorption spectroscopy to elucidate the thermal pericyclic reaction of 1,3-cyclohexadiene (CHD) molecules, i.e., their isomerization to 1,3,5-hexatriene. In the present experimental scheme, the ring-opening reaction is driven by the thermal vibrational energy induced by photoexcitation to the Rydberg states at 6.2 eV and subsequent femtosecond relaxation to the ground state of CHD molecules. The direction of the ring opening, which can be conrotatory or disrotatory, was the primary focus, and the WH rule predicts the disrotatory pathway in the thermal process. We observed the shifts in K-edge absorption of the carbon atom from the 1s orbital to vacant molecular orbitals around 285 eV at a delay between 340 and 600 fs. Furthermore, a theoretical investigation predicts that the shifts depend on the molecular structures along the reaction pathways and the observed shifts in induced absorption are attributed to the structural change in the disrotatory pathway. This confirms that the orbital symmetry is dynamically conserved in the ring-opening reaction of CHD molecules as predicted using the WH rule.

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.

Photochemical Ring-Opening Reaction of 1,3-Cyclohexadiene: Identifying the True Reactive State

The photochemically induced ring-opening isomerization reaction of 1,3-cyclohexadiene to 1,3,5-hexatriene is a textbook example of a pericyclic reaction and has been amply investigated with advanced spectroscopic techniques. The main open question has been the identification of the single reactive state which drives the process. The generally accepted description of the isomerization pathway starts with a valence excitation to the lowest lying bright state, followed by a passage through a conical intersection to the lowest lying doubly excited state, and finally a branching between either the return to the ground state of the cyclic molecule or the actual ring-opening reaction leading to the open-chain isomer. Here, in a joint experimental and computational effort, we demonstrate that the evolution of the excitation-deexcitation process is much more complex than that usually described. In particular, we show that an initially high-lying electronic state smoothly decreasing in energy along the reaction path plays a key role in the ring-opening reaction.

On Electron Pair Rearrangements in Photochemical Reactions: 1,3-Cyclohexadiene Ring Opening

1,3-Cyclohexadiene ring opening has been studied within the bonding evolution theory (BET) framework. We have focused on describing for the first time the electron pair rearrangements leading to the cis-1,3,5-hexatriene (HT) product from CHD. The nature of bonding in this process begins with the weakening of the double bonds in the Franck-Condon region. Along the 1¹B surface, the C-C sigma bond weakens. Meanwhile, its density redistributes toward the whole CHD ring, mainly over double bonds. Breaking of this bond occurs on the 2¹A surface due to the symmetrical splitting of pair density from this region. This density redistributes toward the reaction center once the pericyclic minimum is reached. The formation of the double bonds that characterize HT occurs gradually in the ground state. However, near the 2¹A/1¹A intersection, these bonds are partially established.

Structural or population dynamics: what is revealed by the time-resolved photoelectron spectroscopy of 1,3-cyclohexadiene? A study with an ensemble density functional theory method

Time-resolved photoelectron spectra during the photochemical ring-opening reaction of 1,3-cyclohexadiene (CHD) are modeled by an ensemble density functional theory (eDFT) method. The computational methodology employed in this work is capable of correctly describing the multi-reference effects arising in the ground and excited electronic states of molecules, which is important for the correct description of the ring-opening reaction of CHD. The geometries of molecular species along the non-adiabatic molecular dynamics (NAMD) trajectories reported in a previous study of the CHD photochemical ring-opening were used in this work to calculate the ionization energies and the respective Dyson orbitals for all possible ionization channels. The obtained theoretical time-resolved spectra display decay characteristics in a reasonable agreement with the experimental observations; i.e., the decay (and rise) of the most mechanistically significant signals occurs on the timescale of 100-150 fs. This is very different from the excited state population decay characteristics (τ_S1 = 234 ± 8 fs) obtained in the previous NAMD study. The difference between the population decay and the decay of the photoelectron signal intensity is traced back to the geometric transformation that the molecule undergoes during the photoreaction. This demonstrates the importance of including the geometric information in interpretation of the experimental observations.

Restriction of the conrotatory motion in photo-induced 6π electrocyclic reaction: formation of the excited state of the closed-ring isomer in the cyclization

The electrocyclic reaction dynamics of a photochromic dithiazolylarylene derivative, 2,3-dithiazolylbenzothiophene (DTA) was investigated by using time-resolved transient absorption and fluorescence spectroscopies. The closed-ring isomer of DTA undergoes cycloreversion through the conical intersection mediating the potential energy surfaces of the excited and ground states, which is in agreement with the Woodward–Hoffmann rules for the electrocyclic reactions of 6π electron systems. On the other hand, a large portion of the open-ring isomer undergoes cyclization along the distinct reaction scheme, in which the cyclization takes place in the excited state manifold leading to the formation of the excited state of the closed-ring isomer. The suppression of the geometrical motion of DTA due to the intramolecular interaction could open a new efficient reaction pathway resulting in the formation of the electronically excited state of the product.

Restriction of the molecular geometry opens up a novel pathway in the cyclization reaction of a photochromic dithiazolylarylene derivative.

Excited state dynamics of cis,cis-1,3-cyclooctadiene: Non-adiabatic trajectory surface hopping

We have performed trajectory surface hopping dynamics for cis,cis-1,3-cyclooctadiene to investigate the photochemical pathways involved after being excited to the S₁ state. Our calculations reveal ultrafast decay to the ground state, facilitated by conical intersections involving distortions around the double bonds. The main distortions are localized on one double bond, involving twisting and pyramidalization of one of the carbons of that double bond (similar to ethylene), while a limited number of trajectories decay via delocalized (non-local) twisting of both double bonds. The interplay between local and non-local distortions is important in our understanding of photoisomerization in conjugated systems. The calculations show that a broad range of the conical intersection seam space is accessed during the non-adiabatic events. Several products formed on the ground state have also been observed.

The valence and Rydberg states of dienes

MC-PDFT can give the most accurate excitation energies for both valence and Rydberg states of 1,4-cyclohexadiene and 1,3-cyclohexadiene when the active space includes Rydberg orbitals.

A deep UV trigger for ground-state ring-opening dynamics of 1,3-cyclohexadiene

We explore the photo-induced kinetics of 1,3-cyclohexadiene upon excitation at 200 nm to the 3p state by ultrafast time-resolved, gas-phase x-ray scattering using the Linac Coherent Light Source. Analysis of the scattering anisotropy reveals that the excitation leads to the 3px and 3py Rydberg electronic states, which relax to the ground state with a time constant of 208 ± 11 fs. In contrast to the well-studied 266 nm excitation, at 200 nm the majority of the molecules (76 ± 3%) relax to vibrationally hot cyclohexadiene in the ground electronic state. A subsequent reaction on the ground electronic state surface leads from the hot cyclohexadiene to 1,3,5-hexatriene, with rates for the forward and backward reactions of 174 ± 13 and 355 ± 45 ps, respectively. The scattering pattern of the final hexatriene product reveals a thermal distribution of rotamers about the carbon-carbon single bonds.

Ultrafast ring closing of a diarylethene-based photoswitchable nucleoside

Deoxyuridine nucleosides embodied into diarylethenes form an especial class of photoswitchable compounds that are designed to stack and pair with DNA bases. The molecular geometry can be switched between "open" and "closed" isomers by a pericyclic reaction that affects the stability of the surrounding double helix. This potentially enables light-induced control of DNA hybridization at microscopic resolution. Despite its importance for the optimization of DNA photoswitches, the ultrafast photoisomerization mechanism of these diarylethenes is still not well understood. In this work, femtosecond transient absorption spectroscopy is applied to study the ring closing reaction upon UV excitation with 45 fs pulses. Excited-state absorption decays rapidly and gives rise to the UV-Vis difference spectrum of the "closed" form within ≈15 ps. Time constants of 0.09, 0.49 and 6.6 ps characterize the multimodal dynamics, where a swift recurrence in the signal anisotropy indicates transient population of the intermediate 21A-like state.

Photoinduced C–H bond fission in prototypical organic molecules and radicals

We survey and assess current knowledge regarding the primary photochemistry of hydrocarbon molecules and radicals.

Time-resolved high-harmonic spectroscopy of ultrafast photoisomerization dynamics

We report the first time-resolved high-harmonic spectroscopy (TR-HHS) study of a chemical bond rearrangement. We investigate the transient change of the high-harmonic signal from 1,3-cyclohexadiene (CHD), which undergoes ring-opening and isomerizes to 1,3,5-hexatriene (HT) upon photoexcitation. We associated the harmonic yield variation with the changes in the molecule's electronic state and vibrational frequencies, which are caused by isomerization. This showed us that the electronic excited state of CHD created through two-photon absorption of 3.1 eV photons relaxes almost completely within 100 fs to the electronic ground state of CHD with vibrational excitation. Subsequently, the molecule isomerizes to HT (i.e., ring-opening occurs, around 400 fs after the excitation). The present results demonstrate that TR-HHS, which can track both electronic and nuclear dynamics, is a powerful tool for studying ultrafast photochemical reactions.

Photochemistry of the 1,6-Dideuterio-1,3,5-hexatrienes in Solution: Efficient Terminal Bond Photoisomerization in One-Bond-Twist and Bicycle Pedal Ways

The report that the central bond photoisomerization of the 1,3,5-hexatrienes (Hts) is highly inefficient has encouraged theoreticians to seek conical intersections (CIs) at geometries that can explain rapid nonradiative return to the initially excited isomer. Because they are photochemically silent, torsional relaxations about the terminal double bonds of the Hts have not been evaluated as significant radiationless decay pathways. Study of the photoisomerization of trans,trans,trans- and trans,cis,trans-1,6-dideuterio-1,3,5-hexatrienes ( ttt- and tct-Ht_d2) addresses this issue. Degassed cyclohexane- d₁₂ (C₆D₁₂) and CD₃CN solutions were irradiated at 254 nm in quartz NMR tubes, and the progress of the reactions was followed by ¹H NMR. Photoisomerization rates based on the integration of terminal hydrogen NMR peaks are in reasonable agreement with rates obtained by fitting pure isomer NMR spectra to the phase shift and baseline corrected experimental NMR spectra. The results show that terminal bond isomerization is highly efficient, especially when one considers that central bond isomerization is much more efficient than previously reported and is mainly observed together with terminal bond isomerization. A mechanism involving terminal one-bond-twist (OBT) in competition with a bicycle pedal (BP) process accounts for all terminal and most central bond photoisomerization. OBT central bond isomerization is a minor reaction that is observed primarily in the tct to ttt direction. Most surprising is the prominent role of the BP process in central bond photoisomerization. Proposed initially to account for photoisomerization in free volume constraining media, it is observed here in the absence of medium constraints.

Measurement of Ultrafast Vibrational Coherences in Polyatomic Radical Cations with Strong-Field Adiabatic Ionization

We present a pump-probe method for preparing vibrational coherences in polyatomic radical cations and probing their ultrafast dynamics. By shifting the wavelength of the strong-field ionizing pump pulse from the commonly used 800 nm into the near-infrared (1200-1600 nm), the contribution of adiabatic electron tunneling to the ionization process increases relative to multiphoton absorption. Adiabatic ionization results in predominant population of the ground electronic state of the ion upon electron removal, which effectively prepares a coherent vibrational state ("wave packet") amenable to subsequent excitation. In our experiments, the coherent vibrational dynamics are probed with a weak-field 800 nm pulse and the time-dependent yields of dissociation products measured in a time-of-flight mass spectrometer. We present the measurements on the molecule dimethyl methylphosphonate (DMMP) to illustrate how using 1500 nm pulses for excitation enhances the amplitude of coherent oscillations in ion yields by a factor of 10 as compared to 800 nm pulses. This protocol may be implemented in existing pump-probe setups through the incorporation of an optical parametric amplifier (OPA) for wavelength conversion.

Structural dynamics of photochemical reactions probed by time-resolved photoelectron spectroscopy using high harmonic pulses

Femtosecond ring-opening dynamics of 1,3-cyclohexadiene (CHD) in gas phase upon two-photon excitation at 400 nm (=3.1 eV) was investigated by time-resolved photoelectron spectroscopy using 42 nm (=29.5 eV) high harmonic photons probing the dynamics of the lower-lying occupied molecular orbitals (MOs), which are the fingerprints of the molecular structure. After 500 fs, the photoelectron intensity of the MO constituting the C[double bond, length as m-dash]C sigma bond (σ_{C[double bond, length as m-dash]C}) of CHD was enhanced, while that of the MO forming the C-C sigma bond (σ_CC) of CHD was decreased. The changes in the photoelectron spectra suggest that the ring of CHD opens to form a 1,3,5-hexatriene (HT) after 500 fs. The dynamics of the σ_{C[double bond, length as m-dash]C} and σ_CC bands between 200 and 500 fs reflects the ring deformation to a conical intersection between the 2¹A and 1¹A potential energy surfaces prior to the ring-opening reaction.

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.

Exploring the Dynamics of the Photoinduced Ring-Opening of Heterocyclic Molecules

Excited states formed by electron promotion to an antibonding σ* orbital are now recognized as key to understanding the photofragmentation dynamics of a broad range of heteroatom containing small molecules: alcohols, thiols, amines, and many of their aromatic analogues. Such excited states may be populated by direct photoexcitation, or indirectly by nonadiabatic transfer of population from some other optically excited state (e.g., a ππ* state). This Perspective explores the extent to which the fast-growing literature pertaining to such (n/π)σ*-state mediated bond fissions can inform and enhance our mechanistic understanding of photoinduced ring-opening in heterocyclic molecules.

Molecular Photofragmentation Dynamics in the Gas and Condensed Phases

Exciting a molecule with an ultraviolet photon often leads to bond fission, but the final outcome of the bond cleavage is typically both molecule and phase dependent. The photodissociation of an isolated gas-phase molecule can be viewed as a closed system: Energy and momentum are conserved, and the fragmentation is irreversible. The same is not true in a solution-phase photodissociation process. Solvent interactions may dissipate some of the photoexcitation energy prior to bond fission and will dissipate any excess energy partitioned into the dissociation products. Products that have no analog in the corresponding gas-phase study may arise by, for example, geminate recombination. Here, we illustrate the extent to which dynamical insights from gas-phase studies can inform our understanding of the corresponding solution-phase photochemistry and how, in the specific case of photoinduced ring-opening reactions, solution-phase studies can in some cases reveal dynamical insights more clearly than the corresponding gas-phase study.

Femtosecond x-ray spectroscopy of an electrocyclic ring-opening reaction

The ultrafast light-activated electrocyclic ring-opening reaction of 1,3-cyclohexadiene is a fundamental prototype of photochemical pericyclic reactions. Generally, these reactions are thought to proceed through an intermediate excited-state minimum (the so-called pericyclic minimum), which leads to isomerization via nonadiabatic relaxation to the ground state of the photoproduct. Here, we used femtosecond (fs) soft x-ray spectroscopy near the carbon K-edge (~284 electron volts) on a tabletop apparatus to directly reveal the valence electronic structure of this transient intermediate state. The core-to-valence spectroscopic signature of the pericyclic minimum observed in the experiment was characterized, in combination with time-dependent density functional theory calculations, to reveal overlap and mixing of the frontier valence orbital energy levels. We show that this transient valence electronic structure arises within 60 ± 20 fs after ultraviolet photoexcitation and decays with a time constant of 110 ± 60 fs.

Pulsed Nd:YAG laser-induced photoreaction of cis,cis-1,3-cyclooctadiene at 266 nm: selective cyclization to cis-bicyclo[4.2.0]oct-7-ene

Nanosecond-scale pulsed laser irradiation of cis,cis-1,3-cyclooctadiene via an Nd:YAG laser at 266 nm induced highly selective cyclization of the 1,3-diene moiety to afford cis-bicyclo[4.2.0]oct-7-ene in high yield. The pulsed Nd:YAG laser light is highly monochromatic thereby allowing efficient control of the photoreaction selectivity by controlling the photostationary state.

Ultrafast Time-Resolved Spectroscopy of Diarylethene-Based Photoswitchable Deoxyuridine Nucleosides

Photoswitches based on the diarylethene architecture have been attracting considerable attention for the investigation and control of a variety of biological processes. The reversible photoisomerization reaction between their open- and closed-ring forms can be selectively addressed by irradiation with light of two markedly different wavelengths. In this work, the dynamics of the photochromic ring-opening reaction of four novel diarylethene-based photoswitchable deoxyuridine nucleosides is investigated by femtosecond transient absorption. Upon photoexcitation with sub-20 fs pulses in the first absorption band (500 nm), all four photoswitches showed a fast ballistic excited-state deactivation within less than 500 fs toward the S1/S0 conical intersection. Transient data was globally analyzed, and a relaxation kinetic model with a branching between open and closed ring forms without any loss channels was derived. Ring-opening quantum yields, Φr-o, were found to strongly depend on the substituents (R), ranging from 0.64 (dU(PSI): R = 2-naphthyl) to 0.30 (dU(PSIV): R = 2-pyridyl).

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.

Reactive and unreactive pathways in a photochemical ring opening reaction from 2D femtosecond stimulated Raman

Two-dimensional femtosecond stimulated Raman spectroscopy (2D-FSRS) is used to probe the structural evolution of a modified cyclohexadiene as it undergoes a photoinduced ring opening reaction.

Photoirradiation wavelength dependence of cycloreversion quantum yields of diarylethenes

The cycloreversion quantum yields of compounds 1b and 2b increased upon irradiation with light at shorter wavelengths. The irradiation wavelength dependence is ascribed to the energy barrier on the 2A potential energy surface in the excited electronic state.