Vibrational Motions in Ultrafast Electronic Relaxation of Pyrazine.

Ultrafast internal conversion via a conical intersection is ubiquitous in highly efficient photochemical reactions. Internal conversion from the 1ππ* to the 1nπ* state of pyrazine is the paradigm for this phenomenon; however, the relaxation occurs in such a short time (<20 fs) that the nuclear motion is difficult to observe in real time. The present study precisely measures the vibrational coherence transferred from the 1ππ* state to the 1nπ* state using time-resolved photoelectron spectroscopy with an unprecedented time resolution of 13.3 fs and reveals the key nuclear motions that drive the internal conversion.

(De)localization dynamics of molecular excitons: comparison of mixed quantum-classical and fully quantum treatments.

Molecular excitons play a central role in processes of solar energy conversion, both natural and artificial. It is therefore no wonder that numerous experimental and theoretical investigations in the last decade, employing state-of-the-art spectroscopic techniques and computational methods, have been driven by the common aim to unravel exciton dynamics in multichromophoric systems. Theoretically, exciton (de)localization and transfer dynamics are most often modelled using either mixed quantum-classical approaches (e.g., trajectory surface hopping) or fully quantum mechanical treatments (either using model diabatic Hamiltonians or direct dynamics). Yet, the terms such as "exciton localization" or "exciton transfer" may bear different meanings in different works depending on the method in use (quantum-classical vs. fully quantum). Here, we relate different views on exciton (de)localization. For this purpose, we perform molecular surface hopping simulations on several tetracene dimers differing by a magnitude of exciton coupling and carry out quantum dynamical as well as surface hopping calculations on a relevant model system. The molecular surface hopping simulations are done using efficient long-range corrected time-dependent density functional tight binding electronic structure method, allowing us to gain insight into different regimes of exciton dynamics in the studied systems.

Ultrafast Photoisomerization of Ethylene Studied Using Time-Resolved Extreme Ultraviolet Photoelectron Spectroscopy.

The photoisomerization of isolated ethylene (ethene) was observed in real time from the Franck-Condon region in the 1ππ* state to ground-state products using time-resolved photoelectron spectroscopy with extreme ultraviolet (EUV, 21.7 eV) probe pulses. A combination of filamentation four-wave mixing and high-order harmonic generation was employed to obtain a temporal resolution of 31 ± 2 fs. The nuclear wave packet created by a 160 nm pump pulse accesses C═C twisted geometries within 10 fs, and the population transfer from the excited to the ground state occurs within the next 20-30 fs. Formation of vibrationally highly excited ground-state molecules was observed in less than 45 fs, and they decayed with two time constants of 0.87 and >5 ps. The interpretation of the photoelectron spectra is supported by vertical ionization energies calculated using XMS-CASPT2 along geodesically interpolated reaction paths from the Franck-Condon region to the products.

Ultrafast Ring-Opening Reaction of 1,3-Cyclohexadiene: Identification of Nonadiabatic Pathway via Doubly Excited State.

The photoinduced ring-opening reaction of 1,3-cyclohexadiene (CHD) to produce 1,3,5-hexatriene (HT) plays an essential role in the photobiological synthesis of vitamin D3 in the skin. This reaction follows the Woodward-Hoffmann rule, and C5-C6 bond rupture via an electronically excited state occurs with conrotatory motion of the end CH2 groups. However, it is noted that the photoexcited S1(π,π*) state of CHD is not electronically correlated with the ground state of HT, and the reaction must proceed via nonadiabatic transitions. In the present study, we have clearly observed the nonadiabatic reaction pathway via the doubly excited state of CHD using ultrafast extreme UV photoelectron spectroscopy. The results indicate that the reaction occurs in only 68 fs and creates product vibrational coherence. Extensive computational simulations support the interpretation of experimental results and provide further insights into the electronic dynamics in this paradigmatic electrocyclic ring-opening reaction.

Energetics and optimal molecular packing for singlet fission in BN-doped perylenes: electronic adiabatic state basis screening.

Singlet fission has the potential to increase the efficiency of photovoltaic devices, but the design of suitable chromophores is notoriously difficult. Both the electronic properties of the monomer and the packing motif in the crystal have a big impact on the singlet fission efficiency. Using perylene as an example, it is shown that doping with boron and nitrogen not only helps to align the energy levels but also shifts the stacking position that is optimal for singlet fission. Among all perylene derivatives doped with one or two BN groups, we identify the most suitable isomer for singlet fission with the help of TD-DFT and CASPT2 calculations. The optimal relative disposition of the two monomer units in a cofacially stacked homodimer is explored using two semiempirical models for the singlet fission rate: The first one is the well-known diabatic frontier orbital model, while the second treats singlet fission as a non-adiabatic transition and approximates the rate as the length squared of the non-adiabatic coupling vector between eigenfunctions of the diabatic Hamiltonian.

Effect of varying the TD-lc-DFTB range-separation parameter on charge and energy transfer in a model pentacene/buckminsterfullerene heterojunction.

Atomistic modeling of energy and charge transfer at the heterojunction of organic solar cells is an active field with many remaining outstanding questions owing, in part, to the difficulties in performing reliable photodynamics calculations on very large systems. One approach to being able to overcome these difficulties is to design and apply an appropriate simplified method. Density-functional tight binding (DFTB) has become a popular form of approximate density-functional theory based on a minimal valence basis set and neglect of all but two center integrals. We report the results of our tests of a recent long-range correction (lc) [A. Humeniuk and R. Mitric, J. Chem. Phys. 143, 134120 (2015)] for time-dependent (TD) lc-DFTB by carrying out TD-lc-DFTB fewest switches surface hopping calculations of energy and charge transfer times using the relatively new DFTBABY [A. Humeniuk and R. Mitric, Comput. Phys. Commun. 221, 174 (2017)] program. An advantage of this method is the ability to run enough trajectories to get meaningful ensemble averages. Our interest in the present work is less in determining exact energy and charge transfer rates than in understanding how the results of these calculations vary with the value of the range-separation parameter (Rlc = 1/µ) for a model organic solar cell heterojunction consisting of a gas-phase van der Waals complex P/F made up of a single pentacene (P) molecule together with a single buckminsterfullerene (F) molecule. The default value of Rlc = 3.03 a0 is found to be much too small as neither energy nor charge transfer is observed until Rlc ≈ 10 a0. Tests at a single geometry show that the best agreement with high-quality ab initio spectra is obtained in the limit of no lc (i.e., very large Rlc). A plot of energy and charge transfer rates as a function of Rlc is provided, which suggests that a value of Rlc ≈ 15 a0 yields the typical literature (condensed-phase) charge transfer time of about 100 fs. However, energy and charge transfer times become as high as ∼300 fs for Rlc ≈ 25 a0. A closer examination of the charge transfer process P*/F → P+/F- shows that the initial electron transfer is accompanied by a partial delocalization of the P hole onto F, which then relocalizes back onto P, consistent with a polaron-like picture in which the nuclei relax to stabilize the resultant redistribution of charges.

Size Dependence of Non-Radiative Decay Rates in J-Aggregates.

Fluorophores that emit in the near-infrared (NIR, 700-1700 nm) and have high quantum yields are urgently needed for many technical applications such as organic light-emitting diodes or bioimaging. The design of such chromophores is hampered by the energy gap law, which states that shifting the emission to lower wavelengths is accompanied by a dramatic increase in the nonradiative decay rate. In this article we argue that linear oligomers with J-type excitonic coupling are ideal NIR fluorophores because of the advantageous dependence of the emission energy and the radiative and nonradiative rates on the length N over which the excitation is delocalized. The lowering of the emission energy due to exciton splitting and the linear increase of the radiative rate with length (super-radiance) are well understood. However, less attention has been paid to the decrease of the nonradiative rate with length, which can compensate for the exponential increase due to the energy gap law. According to the exciton model, the Huang-Rhys factors decrease like N-2 while the strength of the nonadiabatic coupling remains approximately constant. Plugging these relations into the Englman-Jortner's energy gap law, we show that for excitonic coupling that is not too strong the nonradiative rate decreases quickly with N. This phenomenon explains the decrease of the nonradiative rate with length in J-aggregates of carbocyanine dyes and the exceptionally high fluorescence quantum yields of linear ethyne-linked zinc-porphyrin arrays, which seemed to defy the energy gap law.

Predicting fluorescence quantum yields for molecules in solution: A critical assessment of the harmonic approximation and the choice of the lineshape function.

For the rational design of new fluorophores, reliable predictions of fluorescence quantum yields from first principles would be of great help. However, efficient computational approaches for predicting transition rates usually assume that the vibrational structure is harmonic. While the harmonic approximation has been used successfully to predict vibrationally resolved spectra and radiative rates, its reliability for non-radiative rates is much more questionable. Since non-adiabatic transitions convert large amounts of electronic energy into vibrational energy, the highly excited final vibrational states deviate greatly from harmonic oscillator eigenfunctions. We employ a time-dependent formalism to compute radiative and non-radiative rates for transitions and study the dependence on model parameters. For several coumarin dyes, we compare different adiabatic and vertical harmonic models (AS, ASF, AH, VG, VGF, and VH), in order to dissect the importance of displacements, frequency changes, and Duschinsky rotations. In addition, we analyze the effect of different broadening functions (Gaussian, Lorentzian, or Voigt). Moreover, to assess the qualitative influence of anharmonicity on the internal conversion rate, we develop a simplified anharmonic model. We address the reliability of these models considering the potential errors introduced by the harmonic approximation and the phenomenological width of the broadening function.

Probing ultrafast dynamics during and after passing through conical intersections.

Time-resolved photoelectron spectroscopy using vacuum-UV probe pulses enables observing ultrafast dynamics during and after passing through conical intersections (CIs). The ring-puckering CI plays a prominent role following the ππ* photoexcitation of furan. More than 90% of the excited molecules safely return to the original ground state, while the remaining 10% transforms into isomers after passing through the puckering CI.

Exciton localization in excited-state dynamics of a tetracene trimer: a surface hopping LC-TDDFTB study.

Excitons in the molecular aggregates of chromophores are key participants in important processes such as photosynthesis or the functioning of organic photovoltaic devices. Therefore, the exploration of exciton dynamics is crucial. Here we report on exciton localization during excited-state dynamics of the recently synthesized tetracene trimer [Liu et al., Org. Lett., 2017, 19, 580]. We employ the surface hopping approach to nonadiabatic molecular dynamics in conjunction with the long-range corrected time-dependent density functional tight binding (LC-TDDFTB) method [Humeniuk and Mitric, Comput. Phys. Commun., 2017, 221, 174]. Utilizing a set of descriptors based on the transition density matrix, we perform comprehensive analysis of exciton dynamics. The obtained results reveal an ultrafast exciton localization to a single tetracene unit of the trimer during excited-state dynamics, along with exciton transfer between units.

Davydov-type excitonic effects on the absorption spectra of parallel-stacked and herringbone aggregates of pentacene: Time-dependent density-functional theory and time-dependent density-functional tight binding.

Exciton formation leads to J-bands in solid pentacene. Describing these exciton bands represents a challenge for both time-dependent (TD) density-functional theory (DFT) and for its semi-empirical analog, namely, for TD density-functional tight binding (DFTB) for three reasons: (i) solid pentacene and pentacene aggregates are bound only by van der Waals forces which are notoriously difficult to describe with DFT and DFTB, (ii) the proper description of the long-range coupling between molecules, needed to describe Davydov splitting, is not easy to include in TD-DFT with traditional functionals and in TD-DFTB, and (iii) mixing may occur between local and charge transfer excitons, which may, in turn, require special functionals. We assess how far TD-DFTB has progressed toward a correct description of this type of exciton by including both a dispersion correction for the ground state and a range-separated hybrid functional for the excited state and comparing the results against corresponding TD-CAM-B3LYP/CAM-B3LYP+D3 results. Analytic results for parallel-stacked ethylene are derived which go beyond Kasha's exciton model [M. Kasha, H. R. Rawls, and A. El-Bayoumi, Pure Appl. Chem. 11, 371 (1965)] in that we are able to make a clear distinction between charge transfer and energy transfer excitons. This is further confirmed when it is shown that range-separated hybrids have a markedly greater effect on charge-transfer excitons than on energy-transfer excitons in the case of parallel-stacked pentacenes. TD-DFT calculations with the CAM-B3LYP functional and TD-lc-DFT calculations lead to negligible excitonic corrections for the herringbone crystal structure, possibly because of an overcorrection of charge-transfer effects (CAM refers to Coulomb attenuated method). In this case, TD-DFT calculations with the B3LYP functional or TD-DFTB calculations parameterized to B3LYP give the best results for excitonic corrections for the herringbone crystal structure as judged from comparison with experimental spectra and with Bethe-Salpeter equation calculations from the literature.

The mechanism of excimer formation: an experimental and theoretical study on the pyrene dimer.

The understanding of excimer formation in organic materials is of fundamental importance, since excimers profoundly influence their functional performance in applications such as light-harvesting, photovoltaics or organic electronics. We present a joint experimental and theoretical study of the ultrafast dynamics of excimer formation in the pyrene dimer in a supersonic jet, which is the archetype of an excimer forming system. We perform simulations of the nonadiabatic photodynamics in the frame of TDDFT that reveal two distinct excimer formation pathways in the gas-phase dimer. The first pathway involves local excited state relaxation close to the initial Franck-Condon geometry that is characterized by a strong excitation of the stacking coordinate exhibiting damped oscillations with a period of 350 fs that persist for several picoseconds. The second excimer forming pathway involves large amplitude oscillations along the parallel shift coordinate with a period of ≈900 fs that after intramolecular vibrational energy redistribution leads to the formation of a perfectly stacked dimer. The electronic relaxation within the excitonic manifold is mediated by the presence of intermolecular conical intersections formed between fully delocalized excitonic states. Such conical intersections may generally arise in stacked π-conjugated aggregates due to the interplay between the long-range and short-range electronic coupling. The simulations are supported by picosecond photoionization experiments in a supersonic jet that provide a time-constant for the excimer formation of around 6-7 ps, in good agreement with theory. Finally, in order to explore how the crystal environment influences the excimer formation dynamics we perform large scale QM/MM nonadiabatic dynamics simulations on a pyrene crystal in the framework of the long-range corrected tight-binding TDDFT. In contrast to the isolated dimer, the excimer formation in the crystal follows a single reaction pathway in which the initially excited parallel slip motion is strongly damped by the interaction with the surrounding molecules leading to the slow excimer stabilization on a picosecond time scale.

Femtosecond dynamics of the 2-methylallyl radical: A computational and experimental study.

We investigate the photodynamics of the 2-methylallyl radical by femtosecond time-resolved photoelectron imaging. The experiments are accompanied by field-induced surface hopping dynamics calculations and the simulation of time-resolved photoelectron intensities and anisotropies, giving insight into the photochemistry and nonradiative relaxation of the radical. 2-methylallyl is excited at 236 nm, 238 nm, and 240.6 nm into a 3p Rydberg state, and the subsequent dynamics is probed by multiphoton ionization using photons of 800 nm. The photoelectron image exhibits a prominent band with considerable anisotropy, which is compatible with the result of theory. The simulations show that the initially excited 3p state is rapidly depopulated to a 3s Rydberg state, from which photoelectrons of high anisotropy are produced. The 3s state then decays within several 100 fs to the D1 (nπ) state, followed by the deactivation of the D1 to the electronic ground state on the ps time scale.

Non-adiabatic dynamics around a conical intersection with surface-hopping coupled coherent states.

A surface-hopping extension of the coupled coherent states-method [D. Shalashilin and M. Child, Chem. Phys. 304, 103-120 (2004)] for simulating non-adiabatic dynamics with quantum effects of the nuclei is put forward. The time-dependent Schrödinger equation for the motion of the nuclei is solved in a moving basis set. The basis set is guided by classical trajectories, which can hop stochastically between different electronic potential energy surfaces. The non-adiabatic transitions are modelled by a modified version of Tully's fewest switches algorithm. The trajectories consist of Gaussians in the phase space of the nuclei (coherent states) combined with amplitudes for an electronic wave function. The time-dependent matrix elements between different coherent states determine the amplitude of each trajectory in the total multistate wave function; the diagonal matrix elements determine the hopping probabilities and gradients. In this way, both interference effects and non-adiabatic transitions can be described in a very compact fashion, leading to the exact solution if convergence with respect to the number of trajectories is achieved and the potential energy surfaces are known globally. The method is tested on a 2D model for a conical intersection [A. Ferretti, J. Chem. Phys. 104, 5517 (1996)], where a nuclear wavepacket encircles the point of degeneracy between two potential energy surfaces and interferes with itself. These interference effects are absent in classical trajectory-based molecular dynamics but can be fully incorpo rated if trajectories are replaced by surface hopping coupled coherent states.

Energy Transfer Between Squaraine Polymer Sections: From Helix to Zigzag and All the Way Back.

We provide a joint experimental and theoretical study of squaraine polymers in solution. The absorption spectra show evidence that two different conformations are present in the polymer: a helix and a zigzag structure. This unique situation allows investigating ultrafast energy-transfer processes between different structural segments within a single polymer chain in solution. The understanding of the underlying dynamics is of fundamental importance for the development of novel materials for light-harvesting and optoelectronic applications. Here, we combine femtosecond transient absorption spectroscopy with time-resolved 2D electronic spectroscopy in order to demonstrate that ultrafast energy transfer within the squaraine polymer chains proceeds from initially excited helix segments to zigzag segments or vice versa, depending on the solvent as well as on the excitation wavenumber. These observations contrast other conjugated polymers such as MEH-PPV where much slower intrachain energy transfer was reported. The reason for the very fast energy transfer in squaraine polymers is most likely a close matching of the density of states between donor and acceptor polymer segments because of the very small reorganization energy in these cyanine-like chromophores.

Long-range correction for tight-binding TD-DFT.

We present two improvements to the tight-binding approximation of time-dependent density functional theory (TD-DFTB): First, we add an exact Hartree-Fock exchange term, which is switched on at large distances, to the ground state Hamiltonian and similarly to the coupling matrix that enters the linear response equations for the calculation of excited electronic states. We show that the excitation energies of charge transfer states are improved relative to the standard approach without the long-range correction by testing the method on a set of molecules from the database in Peach et al. [J. Chem. Phys. 128, 044118 (2008)] which are known to exhibit problematic charge transfer states. The degree of spatial overlap between occupied and virtual orbitals indicates where TD-DFTB and long-range corrected TD-DFTB (lc-TD-DFTB) can be expected to produce large errors. Second, we improve the calculation of oscillator strengths. The transition dipoles are obtained from Slater Koster files for the dipole matrix elements between valence orbitals. In particular, excitations localized on a single atom, which appear dark when using Mulliken transition charges, acquire a more realistic oscillator strength in this way. These extensions pave the way for using lc-TD-DFTB to describe the electronic structure of large chromophoric polymers, where uncorrected TD-DFTB fails to describe the high degree of conjugation and produces spurious low-lying charge transfer states.

Exploring ultrafast dynamics of pyrazine by time-resolved photoelectron imaging.

We present the simulation of time-resolved photoelectron imaging spectra of pyrazine in the gas phase. The approach we have adopted is based on the combination of the ab initio nonadiabatic molecular dynamics "on the fly" with an approximate treatment of the photoionization process using Dyson orbitals and Coulomb functions to describe the bound and ionized states of the photoelectron. The method has been implemented (Humeniuk, A.; et al. J. Chem. Phys 2013, 139, 134104) in the framework of the time-dependent density functional theory and has been applied here to interrogate the ultrafast internal conversion between the S2 and S1 states in pyrazine. Conventional time-resolved photoelectron spectra without angular resolution fail to locate the S2 → S1 internal conversion, because the ionization potentials relevant for the photoionization channels S2 → D1 (π(-1)) and S1 → D0 (n(-1)) are almost identical. Introducing the angular resolution in the photoelectron spectra reveals evidence of such ultrafast internal conversion and provides a more detailed picture of the overall dynamics. The simulated time- and energy-dependent anisotropy map obtained within the Dyson/time-dependent density functional theory approach is in good agreement with its experimental counterpart provided by Horio et al. (Horio, T.; et al. J. Am. Chem. Soc. 2009, 131, 10932). Our theoretical approach represents a general tool for mapping the time- and angle-resolved photoelectron spectra in complex systems and thus can be used to investigate the ultrafast relaxation processes occurring in isolated molecules.

Excited states from quantum Monte Carlo in the basis of Slater determinants.

Building on the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm introduced recently by Booth et al. [J. Chem. Phys. 131, 054106 (2009)] to compute the ground state of correlated many-electron systems, an extension to the computation of excited states (exFCIQMC) is presented. The Hilbert space is divided into a large part consisting of pure Slater determinants and a much smaller orthogonal part (the size of which is controlled by a cut-off threshold), from which the lowest eigenstates can be removed efficiently. In this way, the quantum Monte Carlo algorithm is restricted to the orthogonal complement of the lower excited states and projects out the next highest excited state. Starting from the ground state, higher excited states can be found one after the other. The Schrödinger equation in imaginary time is solved by the same population dynamics as in the ground state algorithm with modified probabilities and matrix elements, for which working formulae are provided. As a proof of principle, the method is applied to lithium hydride in the 3-21G basis set and to the helium dimer in the aug-cc-pVDZ basis set. It is shown to give the correct electronic structure for all bond lengths. Much more testing will be required before the applicability of this method to electron correlation problems of interesting size can be assessed.

Approximate Functionals for Multistate Density Functional Theory.

Recently Lu and Gao [J. Phys. Chem. Lett. 2022, 13 (33), 7762-7769] published a new, rigorous density functional theory for excited states and proved that the projection of the kinetic and electron-repulsion operators into the subspace of the lowest electronic states are a universal functional of the matrix density D(r). This is the first attempt to find an approximation to the multistate universal functional F[D(r)]. It is shown that F (i) does not explicitly depend on the number of electronic states and (ii) is an analytic matrix functional. The Thomas-Fermi-Dirac-von Weizsäcker model and the correlation energy of the homogeneous electron gas are turned into matrix functionals guided by two principles: that each matrix functional should transform properly under basis set transformations and that the ground state functional should be recovered for a single electronic state. Lieb-Oxford-like bounds on the average kinetic and electron-repulsion energies in the subspace are given. When evaluated on the numerically exact matrix density of LiF, this simple approximation reproduces the matrix elements of the electron-repulsion operator in the basis of the exact eigenstates accurately for all bond lengths. In particular the off-diagonal elements of the effective Hamiltonian that come from the interactions of different electronic states can be calculated with the same or better accuracy than the diagonal elements. Unsurprisingly, the largest error comes from the kinetic energy functional. More exact conditions that constrain the functional form of F are needed to go beyond the local density approximation.

Multistate, Polarizable QM/MM Embedding Scheme Based on the Direct Reaction Field Method: Solvatochromic Shifts, Analytical Gradients and Optimizations of Conical Intersections in Solution.

We recently introduced a polarizable embedding scheme based on an integral-exact reformulation of the direct reaction field method (IEDRF) that accounts for the differential solvation of ground and excited states in QM/MM simulations. The polarization and dispersion interactions between the quantum-mechanical (QM) and molecular-mechanical (MM) regions are described by the DRF Hamiltonian, while the Pauli repulsion between explicitly treated QM electrons and the implicit electron density around MM atoms is modeled with effective core potentials. A single Hamiltonian is used for all electronic states so that Born-Oppenheimer states belonging to the same geometry are orthogonal and state crossings are well-defined. In this work, we describe the implementation of the method using graphical processing unit acceleration in TeraChem, where it is combined with multiple electronic structure methods, including Hartree-Fock, time-dependent density functional theory, and complete active space self-consistent field. In contrast with older implementations of the DRF method, integrals of the polarization operators are evaluated exactly. Expressions for ingredients needed to construct analytical gradients and nonadiabatic coupling vectors are derived and tested by optimizing a conical intersection between two excited states in the presence of a polarizable solvent shell. The method is applied to estimate the solvent shifts of absorption energies of a series of donor-acceptor dyes having low-lying charge-transfer states. Even for a nonpolar solvent such as n-hexane, the inclusion of its static polarizability leads to non-negligible shifts that improve the agreement to essentially quantitative levels (0.03 eV) with full-system calculations. Good agreement with the positions of the experimental absorption maxima measured in solution is also observed.