Quantum Mechanical Wave Packet Dynamics at a Conical Intersection with Strong Vibrational Dissipation

Electronic dynamics through conical intersections via non-Markovian stochastic Schrödinger equation with complex modes

Conical intersections (CIs) play a crucial role in photochemical reactions, offering an efficient channel for ultrafast non-adiabatic relaxation of excited states. This significantly influences the reaction pathways and the resulting products. In this work, we utilize the non-Markovian stochastic Schrödinger equation with complex modes method to explore the dynamics of electronic transitions through conical intersections (CIs) in pyrazine. The linear vibronic coupling model serves as the foundational framework, incorporating both intra-state and inter-state electron-vibrational interactions. The dynamics of the excited electronic transitions are analyzed across varying strengths of system-bath coupling and different bath relaxation times. The accuracy of this method is demonstrated by comparing its predictions with those from the hierarchical equations of motion method.

Conical Intersections at Interfaces Revealed by Phase-Cycling Interface-Specific Two-Dimensional Electronic Spectroscopy (i2D-ES)

Conical intersections (CIs) hold significant stake in manipulating and controlling photochemical reaction pathways of molecules at interfaces and surfaces by affecting molecular dynamics therein. Currently, there is no tool for characterizing CIs at interfaces and surfaces. To this end, we have developed phase-cycling interface-specific two-dimensional electronic spectroscopy (i2D-ES) and combined it with advanced computational modeling to explore nonadiabatic CI dynamics of molecules at the air/water interface. Specifically, we integrated the phase locked pump pulse pair with an interface-specific electronic probe to obtain the two-dimensional interface-specific responses. We demonstrate that the nonadiabatic transitions of an interface-active azo dye molecule that occur through the CIs at the interface have different kinetic pathways from those in the bulk water. Upon photoexcitation, two CIs are present: one from an intersection of an optically active S2 state with a dark S1 state and the other from the intersection of the progressed S1 with the ground state S0. We find that the molecular conformations in the ground state are different for interfacial molecules. The interfacial molecules are intimately correlated with the locally populated excited state S2 being farther away from the CI region. This leads to slower nonadiabatic dynamics at the interface than in bulk water. Moreover, we show that the nonadiabatic transition from the S1 dark state to the ground state is significantly longer at the interface than that in the bulk, which is likely due to the orientationally restricted configuration of the excited state at the interface. Our findings suggest that orientational configurations of molecules manipulate reaction pathways at interfaces and surfaces.

Managing temperature in open quantum systems strongly coupled with structured environments

In non-perturbative non-Markovian open quantum systems, reaching either low temperatures with the hierarchical equations of motion (HEOM) or high temperatures with the Thermalized Time Evolving Density Operator with Orthogonal Polynomials Algorithm (T-TEDOPA) formalism in Hilbert space remains challenging. We compare different ways of modeling the environment. Sampling the Fourier transform of the bath correlation function, also called temperature dependent spectral density, proves to be very effective. T-TEDOPA [Tamascelli et al., Phys. Rev. Lett. 123, 090402 (2019)] uses a linear chain of oscillators with positive and negative frequencies, while HEOM is based on the complex poles of an optimized rational decomposition of the temperature dependent spectral density [Xu et al., Phys. Rev. Lett. 129, 230601 (2022)]. Resorting to the poles of the temperature independent spectral density and of the Bose function separately is an alternative when the problem due to the huge number of Bose poles at low temperatures is circumvented. Two examples illustrate the effectiveness of the HEOM and T-TEDOPA approaches: a benchmark pure dephasing case and a two-bath model simulating the dynamics of excited electronic states coupled through a conical intersection. We show the efficiency of T-TEDOPA to simulate dynamics at a finite temperature by using either continuous spectral densities or only all the intramolecular oscillators of a linear vibronic model calibrated from ab initio data of a phenylene ethynylene dimer.

Theoretical model of femtosecond coherence spectroscopy of vibronic excitons in molecular aggregates

When used as pump pulses in transient absorption spectroscopy measurements, femtosecond laser pulses can produce oscillatory signals known as quantum beats. The quantum beats arise from coherent superpositions of the states of the sample and are best studied in the Fourier domain using Femtosecond Coherence Spectroscopy (FCS), which consists of one-dimensional amplitude and phase plots of a specified oscillation frequency as a function of the detection frequency. Prior works have shown ubiquitous amplitude nodes and π phase shifts in FCS from excited-state vibrational wavepackets in monomer samples. However, the FCS arising from vibronic-exciton states in molecular aggregates have not been studied theoretically. Here, we use a model of vibronic-exciton states in molecular dimers based on displaced harmonic oscillators to simulate FCS for dimers in two important cases. Simulations reveal distinct spectral signatures of excited-state vibronic-exciton coherences in molecular dimers that may be used to distinguish them from monomer vibrational coherences. A salient result is that, for certain relative orientations of the transition dipoles, the key resonance condition between the electronic coupling and the frequency of the vibrational mode may yield strong enhancement of the quantum-beat amplitude and, perhaps, also cause a significant decrease of the oscillation frequency to a value far lower than the vibrational frequency. Future studies using these results will lead to new insights into the excited-state coherences generated in photosynthetic pigment-protein complexes.

Spin-vibronic coherence drives singlet-triplet conversion

Design-specific control over the transitions between excited electronic states with different spin multiplicities is of the utmost importance in molecular and materials chemistry1-3. Previous studies have indicated that the combination of spin-orbit and vibronic effects, collectively termed the spin-vibronic effect, can accelerate quantum-mechanically forbidden transitions at non-adiabatic crossings4,5. However, it has been difficult to identify precise experimental manifestations of the spin-vibronic mechanism. Here we present coherence spectroscopy experiments that reveal the interplay between the spin, electronic and vibrational degrees of freedom that drive efficient singlet-triplet conversion in four structurally related dinuclear Pt(II) metal-metal-to-ligand charge-transfer (MMLCT) complexes. Photoexcitation activates the formation of a Pt-Pt bond, launching a stretching vibrational wavepacket. The molecular-structure-dependent decoherence and recoherence dynamics of this wavepacket resolve the spin-vibronic mechanism. We find that vectorial motion along the Pt-Pt stretching coordinates tunes the singlet and intermediate-state energy gap irreversibly towards the conical intersection and subsequently drives formation of the lowest stable triplet state in a ratcheting fashion. This work demonstrates the viability of using vibronic coherences as probes6-9 to clarify the interplay among spin, electronic and nuclear dynamics in spin-conversion processes, and this could inspire new modular designs to tailor the properties of excited states.

Vibrational response functions for multidimensional electronic spectroscopy in nonadiabatic models

The interplay of nuclear and electronic dynamics characterizes the multidimensional electronic spectra of various molecular and solid-state systems. Theoretically, the observable effect of such interplay can be accounted for by response functions. Here, we report analytical expressions for the response functions corresponding to a class of model systems. These are characterized by coupling between the diabatic electronic states and the vibrational degrees of freedom, resulting in linear displacements of the corresponding harmonic oscillators, and by nonadiabatic couplings between pairs of diabatic states. In order to derive the linear response functions, we first perform the Dyson expansion of the relevant propagators with respect to the nonadiabatic component of the Hamiltonian, then derive and expand with respect to the displacements the propagators at given interaction times, and finally provide analytical expressions for the time integrals that lead to the different contributions to the linear response function. The approach is then applied to the derivation of third-order response functions describing different physical processes: ground state bleaching, stimulated emission, excited state absorption, and double quantum coherence. Comparisons between the results obtained up to sixth order in the Dyson expansion and independent numerical calculation of the response functions provide evidence of the series convergence in a few representative cases.

Direct Monitoring of Conical Intersection Passage via Electronic Coherences in Twisted X-Ray Diffraction

Quantum coherences in electronic motions play a critical role in determining the pathways and outcomes of virtually all photophysical and photochemical molecular processes. However, the direct observation of electronic coherences in the vicinity of conical intersections remains a formidable challenge. We propose a novel time-resolved twisted x-ray diffraction technique that can directly monitor the electronic coherences created as the molecule passes through a conical intersection. We show that the contribution of electronic populations to this signal is canceled out when using twisted x-ray beams that carry a light orbital angular momentum, providing a direct measurement of transient electronic coherences in gas-phase molecules.

Vibrational response functions for multidimensional electronic spectroscopy in the adiabatic regime: A coherent-state approach

Multi-dimensional spectroscopy represents a particularly insightful tool for investigating the interplay of nuclear and electronic dynamics, which plays an important role in a number of photophysical processes and photochemical reactions. Here, we present a coherent state representation of the vibronic dynamics and of the resulting response functions for the widely used linearly displaced harmonic oscillator model. Analytical expressions are initially derived for the case of third-order response functions in an N-level system, with ground state initialization of the oscillator (zero-temperature limit). The results are then generalized to the case of Mth order response functions, with arbitrary M. The formal derivation is translated into a simple recipe, whereby the explicit analytical expressions of the response functions can be derived directly from the Feynman diagrams. We further generalize to the whole set of initial coherent states, which form an overcomplete basis. This allows one, in principle, to derive the dependence of the response functions on arbitrary initial states of the vibrational modes and is here applied to the case of thermal states. Finally, a non-Hermitian Hamiltonian approach is used to include in the above expressions the effect of vibrational relaxation.

Probing nonadiabatic dynamics with attosecond pulse trains and soft x-ray Raman spectroscopy

Linear off-resonant x-ray Raman techniques are capable of detecting the ultrafast electronic coherences generated when a photoexcited wave packet passes through a conical intersection. A hybrid femtosecond or attosecond probe pulse is employed to excite the system and stimulate the emission of the signal photon, where both fields are components of a hybrid pulse scheme. In this paper, we investigate how attosecond pulse trains, as provided by high-harmonic generation processes, perform as probe pulses in the framework of this spectroscopic technique, instead of single Gaussian pulses. We explore different combination schemes for the probe pulse as well as the impact of parameters of the pulse trains on the signals. Furthermore, we show how Raman selection rules and symmetry consideration affect the spectroscopic signal, and we discuss the importance of vibrational contributions to the overall signal. We use two different model systems, representing molecules of different symmetries, and quantum dynamics simulations to study the difference in the spectra. The results suggest that such pulse trains are well suited to capture the key features associated with the electronic coherence.

Diffractive Imaging of Conical Intersections Amplified by Resonant Infrared Fields

The fate of virtually all photochemical reactions is determined by conical intersections. These are energetically degenerate regions of molecular potential energy surfaces that strongly couple electronic states, thereby enabling fast relaxation channels. Their direct spectroscopic detection relies on weak features that are often buried beneath stronger, less interesting contributions. For azobenzene photoisomerization, a textbook photochemical reaction, we demonstrate how a resonant infrared field can be employed during the conical intersection passage to significantly enhance its coherence signatures in time-resolved X-ray diffraction while leaving the product yield intact. This transition-state amplification holds promise to bring signals of conical intersections above the detection threshold.

Funneling dynamics in a phenylacetylene trimer: Coherent excitation of donor excitonic states and their superposition

Funneling dynamics in conjugated dendrimers has raised great interest in the context of artificial light-harvesting processes. Photoinduced relaxation has been explored by time-resolved spectroscopy and simulations, mainly by semiclassical approaches or referring to open quantum systems methods, within the Redfield approximation. Here, we take the benefit of an ab initio investigation of a phenylacetylene trimer, and in the spirit of a divide-and-conquer approach, we focus on the early dynamics of the hierarchy of interactions. We build a simplified but realistic model by retaining only bright electronic states and selecting the vibrational domain expected to play the dominant role for timescales shorter than 500 fs. We specifically analyze the role of the in-plane high-frequency skeletal vibrational modes involving the triple bonds. Open quantum system non-adiabatic dynamics involving conical intersections is conducted by separating the electronic subsystem from the high-frequency tuning and coupling vibrational baths. This partition is implemented within a robust non-perturbative and non-Markovian method, here the hierarchical equations of motion. We will more precisely analyze the coherent preparation of donor states or of their superposition by short laser pulses with different polarizations. In particular, we extend the π-pulse strategy for the creation of a superposition to a V-type system. We study the relaxation induced by the high-frequency vibrational collective modes and the transitory dissymmetry, which results from the creation of a superposition of electronic donor states.

Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

Two-dimensional electronic-vibrational spectroscopic study of conical intersection dynamics: an experimental and electronic structure study

The relaxation from the lowest singlet excited state of the triphenylmethane dyes, crystal violet and malachite green, is studied via two-dimensional electronic-vibrational (2DEV) spectroscopy.

Geometric phase effects on photodissociation dynamics of diatomics

We investigated the effect of the geometric phase (GP) on photodissociation dynamics at a light-induced conical intersection (LICI) through exact quantum dynamical calculations. By taking the one-photon photodissociation of H 2 + ionic molecules as an example, we explored the conditions wherein the LICI associated GP affects dissociation dynamics. We found that GP leads to a phase shift between the angular distributions of GP included and GP excluded photofragments. This effect is more pronounced when the energy of the initial vibrational level is above the energy of the LICI point.

Simulation of Quantum Dynamics of Excitonic Systems at Finite Temperature: an efficient method based on Thermo Field Dynamics

Quantum electron-vibrational dynamics in molecular systems at finite temperature is described using an approach based on Thermo Field Dynamics theory. This formulation treats temperature effects in the Hilbert space without introducing the Liouville space. The solution of Thermo Field Dynamics equations with a novel technique for the propagation of Tensor Trains (Matrix Product States) is implemented and discussed. The methodology is applied to the study of the exciton dynamics in the Fenna-Mathews-Olsen complex using a realistic structured spectral density to model the electron-phonon interaction. The results of the simulations highlight the effect of specific vibrational modes on the exciton dynamics and energy transfer process, as well as call for careful modeling of electron-phonon couplings.

Intrinsic and light-induced nonadiabatic phenomena in the NaI molecule

Nonadiabatic effects play a very important role in controlling chemical dynamical processes. They are strongly related to avoided crossings (AC) or conical intersections (CIs) which can either be present naturally or induced by classical laser light in a molecular system. The latter are named as "light-induced avoided crossings" (LIACs) and "light-induced conical intersections" (LICIs). By performing one or two dimensional quantum dynamical calculations LIAC and LICI situations can easily be created even in diatomic molecules. Applying such calculations for the NaI molecule, which is a strongly coupled diatomic in field free case, significant differences between the impact of the LIAC and LICI on the ground state population dynamics were observed. Moreover, obtained results undoubtedly demonstrate that the effect of the LIAC and LICI on the dynamics strongly depends on the intensity and the frequency of the applied laser field as well as the permanent dipole moments of the molecule.

On the Dynamics through a Conical Intersection

Conical intersections represent critical topological features of potential energy surfaces and open ultrafast nonradiative deactivation channels for photoexcited molecules. In the following, we investigate how this funneling picture is transposed in the eyes of the exact factorization formalism for a 2D model system. The exact factorization of the total molecular wave function leads to the fundamental concept of time-dependent potential energy surface and time-dependent vector potential, whose behavior during a dynamics through a conical intersection has up to now remained unexplored. Despite the fact that these quantities might be viewed as time-dependent generalizations of the adiabatic potential energy surfaces and the nonadiabatic coupling vectors, characteristic quantities appearing in the Born-Oppenheimer framework, we observe that they do not exhibit particular topological features in the region of conical intersection but still reflect the complex dynamics of the nuclear wavepacket.

Stimulated Raman Scattering: From Bulk to Nano

Stimulated Raman scattering (SRS) describes a family of techniques first discovered and developed in the 1960s. Whereas the nascent history of the technique is parallel to that of laser light sources, recent advances have spurred a resurgence in its use and development that has spanned across scientific fields and spatial scales. SRS is a nonlinear technique that probes the same vibrational modes of molecules that are seen in spontaneous Raman scattering. While spontaneous Raman scattering is an incoherent technique, SRS is a coherent process, and this fact provides several advantages over conventional Raman techniques, among which are much stronger signals and the ability to time-resolve the vibrational motions. Technological improvements in pulse generation and detection strategies have allowed SRS to probe increasingly smaller volumes and shorter time scales. This has enabled SRS research to move from its original domain, of probing bulk media, to imaging biological tissues and single cells at the micro scale, and, ultimately, to characterizing samples with subdiffraction resolution at the nanoscale. In this Review, we give an overview of the history of the technique, outline its basic properties, and present historical and current uses at multiple length scales to underline the utility of SRS to the molecular sciences.

Coupled HOOP signature correlates with quantum yield of isorhodopsin and analog pigments

With a quantum yield of 0.66±0.03 the photoisomerization efficiency of the visual pigment rhodopsin (11-cis⇒all-trans chromophore) is exceptionally high. This is currently explained by coherent coupling of the excited state electronic wavepacket with local vibrational nuclear modes, facilitating efficient cross-over at a conical intersection onto the photoproduct energy surface. The 9-cis counterpart of rhodopsin, dubbed isorhodopsin, has a much lower quantum yield (0.26±0.03), which, however, can be markedly enhanced by modification of the retinal chromophore (7,8-dihydro and 9-cyclopropyl derivatives). The coherent coupling in the excited state is promoted by torsional skeletal and coupled HOOP vibrational modes, in combination with a twisted conformation around the isomerization region. Since such torsion will strongly enhance the infrared intensity of coupled HOOP modes, we investigated FTIR difference spectra of rhodopsin, isorhodopsin and several analog pigments in the spectral range of isolated and coupled HCCH wags. As a result we propose that the coupled HOOP signature in these retinal pigments correlates with the distribution of torsion over counteracting segments in the retinylidene polyene chain. As such the HOOP signature can act as an indicator for the photoisomerization efficiency, and can explain the higher quantum yield of the 7,8-dihydro and 9-cyclopropyl-isorhodopsin analogs.

Probing the Interstate Coupling near a Conical Intersection by Optical Spectroscopy

Conical intersections are points where adiabatic potential energy surfaces cross. The interstate coupling between the potential energy surfaces plays a crucial role in many processes associated with conical intersections. Still no method exists to measure this coupling driving the chemical reactions between the potential energy surfaces involved. In this Letter, using a generic model for photoisomerization, we propose a novel experimental approach to estimate the coupling that mixes the electronic states near a conical intersection. The approach is based on analyzing the vibrational wavepacket of the reactant in the adiabatic ground and excited electronic states. The nuclear wavepacket dynamics are extracted from linear absorption and two-dimensional electronic spectroscopy. Comparing the frequencies of the coupling mode in the adiabatic ground and excited states from models with and without coupling between the potential energy surfaces suggests an experimental tool to determine the interstate coupling.

Impact of Vibrational Coherence on the Quantum Yield at a Conical Intersection

We study the vibrationally coherent quantum dynamics of an electronic wave packet in the vicinity of a conical intersection within a three-state two-mode model. By transforming the coherent tuning and coupling modes into the bath, the underdamped dynamics of the resulting effective three-state model is solved efficiently by the numerically exact hierarchy equation of motion approach. The transient excited-state absorption and two-dimensional spectra reveal the impact of vibrational coherence on the relaxation pathways of the wave packet. We find that both the quantum yield and the isomerization rate are crucially influenced by the vibrational coherence of the wave packet. A less coherent wave packet can traverse the conical intersection more rapidly, while the resulting quantum yield is smaller. Finally, we show that repeated passages of the wave packet through the conical intersection can lead to measurable interference effects in the form of Stueckelberg oscillations.

Dissipative dynamics at conical intersections: simulations with the hierarchy equations of motion method

The effect of a dissipative environment on the ultrafast nonadiabatic dynamics at conical intersections is analyzed for a two-state two-mode model chosen to represent the S₂(ππ*)–S₁(nπ*) conical intersection in pyrazine (the system) which is bilinearly coupled to infinitely many harmonic oscillators in thermal equilibrium (the bath). The system–bath coupling is modeled by the Drude spectral function. The equation of motion for the reduced density matrix of the system is solved numerically exactly with the hierarchy equation of motion method using graphics-processor-unit (GPU) technology. The simulations are valid for arbitrary strength of the system–bath coupling and arbitrary bath memory relaxation time. The present computational studies overcome the limitations of weak system–bath coupling and short memory relaxation time inherent in previous simulations based on multi-level Redfield theory [A. Kühl and W. Domcke, J. Chem. Phys. 2002, 116, 263]. Time evolutions of electronic state populations and time-dependent reduced probability densities of the coupling and tuning modes of the conical intersection have been obtained. It is found that even weak coupling to the bath effectively suppresses the irregular fluctuations of the electronic populations of the isolated two-mode conical intersection. While the population of the upper adiabatic electronic state (S₂) is very efficiently quenched by the system–bath coupling, the population of the diabatic ππ* electronic state exhibits long-lived oscillations driven by coherent motion of the tuning mode. Counterintuitively, the coupling to the bath can lead to an enhanced lifetime of the coherence of the tuning mode as a result of effective damping of the highly excited coupling mode, which reduces the strong mode–mode coupling inherent to the conical intersection. The present results extend previous studies of the dissipative dynamics at conical intersections to the nonperturbative regime of system–bath coupling. They pave the way for future first-principles simulations of femtosecond time-resolved four-wave-mixing spectra of chromophores in condensed phases which are nonperturbative in the system dynamics, the system–bath coupling as well as the field-matter coupling.