From Surface Hopping to Quantum Dynamics and Back. Finding Essential Electronic and Nuclear Degrees of Freedom and Optimal Surface Hopping Parameters

From Lithium and Sodium Superoxides to Singlet-Oxygen - Insights into the Mechanism of Dissociation Using SHARC-MD

The formation of highly reactive singlet oxygen from alkaline superoxides presents an important reactivity of this component class. Investigations of the reaction paths such as disproportionation of LiO2 and NaO2 have been presented. Furthermore, the dissociation of these superoxide systems have been discussed as an alternative reaction channel that also allows the formation of singlet oxygen. Here, we present a fundamental study of the electronic nature and dissociation behaviour of the alkali superoxides. The molecular systems were calculated at the CASSCF/CASPT2-level of theory. We determined the minimum energy crossing points along the dissociation required to form triplet oxygen 3O2 and singlet oxygen 1O2. Building on these results, a surface-hopping AIMD-simulation was performed employing the SHARC program package to follow the electronic transitions along the minimum energy crossing points during the dissociation. The feasibility of populating the electronic state corresponding to the formation of singlet oxygen during dissociation was demonstrated. For LiO2, 6.85 % of the trajectories were found to terminate under formation of 1O2, whereas for NaO2 only 1.68 % of the trajectories ended up in 1O2 formation. This represents an inverse trend to that reported in the literature. This observation suggests that the dissociation is a viable, monomolecular reaction path to 1O2 that complements the disproportionation pathway.

Photophysics and photochemistry of (n-Bu4N)2[Pt(NO3)6] in acetonitrile: ultrafast pump-probe spectroscopy and quantum chemical insight

The ultrafast processes caused by photoexcitation of (n-Bu4N)2[Pt(NO3)6] complex in acetonitrile were studied by means of transient absorption (TA) pump-probe spectroscopy and verified by quantum chemical calculations. The primary photochemical process was found to be an inner-sphere electron transfer followed by an escape of an •NO3 radical to the bulk solution. The reaction occurs via the dissociative triplet excited LMCT state of the initial complex. Based on the experimental data and quantum chemical calculations, the mechanism of ultrafast photophysical and photochemical processes is proposed.

Simulating Attochemistry: Which Dynamics Method to Use?

Attochemistry aims to exploit the properties of coherent electronic wavepackets excited via attosecond pulses to control the formation of photoproducts. Such molecular processes can, in principle, be simulated with various nonadiabatic dynamics methods, yet the impact of the approximations underlying the methods is rarely assessed. The performances of widely used mixed quantum-classical approaches, Tully surface hopping, and classical Ehrenfest methods are evaluated against the high-accuracy DD-vMCG quantum dynamics. This comparison is conducted for the valence ionization of fluorobenzene. Analyzing the nuclear motion induced in the branching space of the nearby conical intersection, the results show that the mixed quantum-classical methods reproduce quantitatively the average motion of a quantum wavepacket when initiated on a single electronic state. However, they fail to properly capture the nuclear motion induced by an electronic wavepacket along the derivative coupling, the latter originating from the quantum electronic coherence property, key to attochemistry.

Photochemistry of (n-Bu4N)2[Pt(NO3)6] in acetonitrile

Photochemistry of the (n-Bu4N)2[Pt(NO3)6] complex in acetonitrile was studied by means of stationary photolysis and nanosecond laser flash photolysis. The primary photochemical process was found to be an intramolecular electron transfer followed by an escape of an •NO3 radical to the solution bulk. The spectra of two successive Pt(III) intermediates were detected in the microsecond time domain, and their spectral and kinetic characteristics were determined. These intermediates were identified as PtIII(NO3)52- and PtIII(NO3)4- complexes. Disproportionation of Pt(III) species resulted in formation of final Pt(II) products.

Excitation energy transfer and vibronic relaxation through light-harvesting dendrimer building blocks: A nonadiabatic perspective

The light-harvesting excitonic properties of poly(phenylene ethynylene) (PPE) extended dendrimers (tree-like π-conjugated macromolecules) involve a directional cascade of local excitation energy transfer (EET) processes occurring from the "leaves" (shortest branches) to the "trunk" (longest branch), which can be viewed from a vibronic perspective as a sequence of internal conversions occurring among a connected graph of nonadiabatically coupled locally excited electronic states via conical intersections. The smallest PPE building block that is able to exhibit EET, the asymmetrically meta-substituted PPE oligomer with one acetylenic bond on one side and two parallel ones on the other side (hence, 2-ring and 3-ring para-substituted pseudo-fragments), is a prototype and the focus of the present work. From linear-response time-dependent density functional theory electronic-structure calculations of the molecule as regards its first two nonadiabatically coupled, optically active, singlet excited states, we built a (1 + 2)-state-8-dimensional vibronic-coupling Hamiltonian model for running subsequent multiconfiguration time-dependent Hartree wavepacket relaxations and propagations, yielding both steady-state absorption and emission spectra as well as real-time dynamics. The EET process from the shortest branch to the longest one occurs quite efficiently (about 80% quantum yield) within the first 25 fs after light excitation and is mediated vibrationally through acetylenic and quinoidal bond-stretching modes together with a particular role given to the central-ring anti-quinoidal rock-bending mode. Electronic and vibrational energy relaxations, together with redistributions of quantum populations and coherences, are interpreted herein through the lens of a nonadiabatic perspective, showing some interesting segregation among the foremost photoactive degrees of freedom as regards spectroscopy and reactivity.

Coronene: a model for ultrafast dynamics in graphene nanoflakes and PAHs

Assuming a delta pulse excitation, quantum wavepackets are propagated on the excited state manifold in the energy range from 3.4-5.0 eV for coronene and 2.4-3.5 eV for circumcoronene to study the time evolution of the states as well as their lifetimes. The full-dimensional (102 and 210 degrees of freedom for coronene and circumcoronene respectively) non-adiabatic dynamics simulated with the ML-MCTDH method on twelve coupled singlet electronic states show that the different absorption spectra are only due to electronic delocalisation effects that change the excited state energies, but the structural dynamics in both compounds are identical. Breathing and tilting motions drive the decay dynamics of the electronic states away from the Frank-Condon region independently of the size of the aromatic system. This promising result allows the use of coronene as a model system for the dynamics of larger polycyclic aromatic hydrocarbons (PAHs) and graphene one dimensional sheets or nanoflakes.

Vibrational Coherences of the Photoinduced Mixed-Valent Creutz-Taube Ion Revealed by Excited State Dynamics

A recent study of photoinduced mixed-valency in the one-electron reduced form (μ-pz)[RuII(NH3)5]24+ of the Creutz-Taube ion used transient absorption spectroscopy with vis-NIR broadband detection to uncover a mixed-valent excited state with a typical intervalence charge transfer band and a nanosecond lifetime [Pieslinger et al. Angew. Chem., Int. Ed. 2022, 61, e202211747]. Herein, we use excited state dynamics simulations with implicit solvation to elucidate the electronic and vibrational evolution in the first 10 ps after the optical excitation. A manifold of excited states with weak interaction between the metal centers is populated already at time zero due to the breakdown of the Condon approximation and dominates the population of electronic states at short time scales (<0.5 ps). A long-lived vibrational wave packet mostly confined to oscillations of the metal center-bridge distances is observed. The oscillations are traced to the electronic structure properties of states with weak metal-metal coupling. The long-lived mixed-valent excited state of the Creutz-Taube ion analogue is formed vibrationally cold and has a more compact geometry. While experimentally, intersystem crossing and vibrational relaxation were deduced to be completed within 1 ps, our analysis indicates that both processes might persist at longer times.

Vibrational Funnels for Energy Transfer in Organic Chromophores

Photoinduced intramolecular energy transfers in multichromophoric molecules involve nonadiabatic vibronic channels that act as energy transfer funnels. They commonly take place through specific directions of motion dictated by the nonadiabatic coupling vectors. Vibrational funnels may support persistent coherences between electronic states and sometimes delineate the presence of minor alternative energy transfer pathways. The ultimate confirmation of their role on the interchromophoric energy transfer can be achieved by performing nonadiabatic excited-state molecular dynamics simulations by selectively freezing the nuclear motions in question. Our results point out this strategy as a useful tool to identify and evaluate the impact of these vibrational funnels on the energy transfer processes and guide the in silico design of materials with tunable properties and enhanced functionalities. Our work encourages applications of this methodology to different chemical and biochemical processes such as reactive scattering and protein conformational changes, to name a few.

Mind the GAP: quantifying the breakdown of the linear vibronic coupling Hamiltonian

Excited state dynamics play a critical role across a broad range of scientific fields. Importantly, the highly non-equilibrium nature of the states generated by photoexcitation means that excited state simulations should usually include an accurate description of the coupled electronic-nuclear motion, which often requires solving the time-dependent Schrödinger equation (TDSE). One of the biggest challenges for these simulations is the requirement to calculate the PES over which the nuclei evolve. An effective approach for addressing this challenge is to use the approximate linear vibronic coupling (LVC) Hamiltonian, which enables a model potential to be parameterised using relatively few quantum chemistry calculations. However, this approach is only valid provided there are no large amplitude motions in the excited state dynamics. In this paper we introduce and deploy a metric, the global anharmonicity parameter (GAP), which can be used to assess the accuracy of an LVC potential. Following its derivation, we illustrate its utility by applying it to three molecules exhibiting different rigidity in their excited states.

Trend in light-induced excited-state spin trapping in Fe(II)-based spin crossover systems

A computational study of the light-induced excited spin-state trapping (LIESST) in a number of Fe(II) spin crossover complexes, coordinated by monodentate, bidentate and multidentate ligands is carried out, with the goal to uncover the trend in the low temperature relaxation rate. A nine order of magnitude change in low temperature relaxation rate is observed among the complexes. The trend is rationalized in terms of the change in metal-ligand covalency, numerically estimated by the crystal orbital Hamiltonian population, thus influencing the back donation or delocalization of the electrons from the low-lying Fe(II)-centered molecular orbital to the empty low-lying ligand-centered π* antibonding molecular orbitals.

Vibronic Photoexcitation Dynamics of Perylene Diimide: Computational Insights

Perylene diimide (PDI) represents a prototype material for organic optoelectronic devices because of its strong optical absorbance, chemical stability, efficient energy transfer, and optical and chemical tunability. Herein, we analyze in detail the vibronic relaxation of its photoexcitation using nonadiabatic excited-state molecular dynamics simulations. We find that after the absorption of a photon, which excites the electron to the second excited state, S₂, induced vibronic dynamics features persistent modulations in the spatial localization of electronic and vibrational excitations. These energy exchanges are dictated by strong vibronic couplings that overcome structural disorders and thermal fluctuations. Specifically, the electronic wavefunction periodically swaps between localizations on the right and left sides of the molecule. Within 1 ps of such dynamics, a nonradiative transition to the lowest electronic state, S₁, takes place, resulting in a complete delocalization of the wavefunction. The observed vibronic dynamics emerges following the electronic energy deposition in the direction that excites a combination of two dominant vibrational normal modes. This behavior is maintained even with a chemical substitution that breaks the symmetry of the molecule. We believe that our findings elucidate the nature of the complex dynamics of the optically excited states and, therefore, contribute to the development of tunable functionalities of PDIs and their derivatives.

Surface Hopping Dynamics on Vibronic Coupling Models

The simulation of photoinduced non-adiabatic dynamics is of great relevance in many scientific disciplines, ranging from physics and materials science to chemistry and biology. Upon light irradiation, different relaxation processes take place in which electronic and nuclear motion are intimately coupled. These are best described by the time-dependent molecular Schrödinger equation, but its solution poses fundamental practical challenges to contemporary theoretical chemistry. Two widely used and complementary approaches to this problem are multiconfigurational time-dependent Hartree (MCTDH) and trajectory surface hopping (SH). MCTDH is an accurate fully quantum-mechanical technique but often is feasible only in reduced dimensionality, in combination with approximate vibronic coupling (VC) Hamiltonians, or both (i.e., reduced-dimensional VC potentials). In contrast, SH is a quantum–classical technique that neglects most nuclear quantum effects but allows nuclear dynamics in full dimensionality by calculating potential energy surfaces on the fly. If nuclear quantum effects do not play a central role and a linear VC (LVC) Hamiltonian is appropriate—e.g., for stiff molecules that generally keep their conformation in the excited state—then it seems advantageous to combine the efficient LVC and SH techniques. In this Account, we describe how surface hopping based on an LVC Hamiltonian (SH/LVC)—as recently implemented in the SHARC surface hopping package—can provide an economical and automated approach to simulate non-adiabatic dynamics. First, we illustrate the potential of SH/LVC in a number of showcases, including intersystem crossing in SO₂, intra-Rydberg dynamics in acetone, and several photophysical studies on large transition-metal complexes, which would be much more demanding or impossible to perform with other methods. While all of the applications provide very useful insights into light-induced phenomena, they also hint at difficulties faced by the SH/LVC methodology that need to be addressed in the future. Second, we contend that the SH/LVC approach can be useful to benchmark SH itself. By the use of the same (LVC) potentials as MCTDH calculations have employed for decades and by relying on the efficiency of SH/LVC, it is possible to directly compare multiple SH test calculations with a MCTDH reference and ponder the accuracy of various correction algorithms behind the SH methodology, such as decoherence corrections or momentum rescaling schemes. Third, we demonstrate how the efficiency of SH/LVC can also be exploited to identify essential nuclear and electronic degrees of freedom to be employed in more accurate MCTDH calculations. Lastly, we show that SH/LVC is able to advance the development of SH protocols that can describe nuclear dynamics including explicit laser fields—a very challenging endeavor for trajectory-based schemes. To end, this Account compiles the typical costs of contemporary SH simulations, evidencing the great advantages of using parametrized potentials. The LVC model is a sleeping beauty that, kissed by SH, is fueling the field of excited-state molecular dynamics. We hope that this Account will stimulate future research in this direction, leveraging the advantages of the SH/VC schemes to larger extents and extending their applicability to uncharted territories.

Photoinduced Low-Spin → High-Spin Mechanism of an Octahedral Fe(II) Complex Revealed by Synergistic Spin-Vibronic Dynamics

The Fe(II) low-spin (LS; ¹A_1g, t_2g⁶e_g⁰) → high-spin (HS; ⁵T_2g, t_2g⁴e_g²) light-induced excited spin state trapping (LIESST) mechanism solely involving metal-centered states is revealed by synergistic spin-vibronic dynamics simulations. For the octahedral [Fe(NCH)₆]²⁺ complex, we identify an initial ∼100 fs ¹T_1g → ³T_2g intersystem crossing, controlled by vibronic coupling to antisymmetric Fe-N stretching motion. Subsequently, population branching into ³T_1g, ⁵T_2g (HS), and ¹A_1g (LS) is observed on a subpicosecond time scale, with the dynamics dominated by coherent Fe-N breathing wavepackets. These findings are consistent with ultrafast experiments, methodologically establish a new state of the art, and will give a strong impetus for further intriguing dynamical studies on LS → HS photoswitching.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

The Quest to Simulate Excited-State Dynamics of Transition Metal Complexes

This Perspective describes current computational efforts in the field of simulating photodynamics of transition metal complexes. We present the typical workflows and feature the strengths and limitations of the different contemporary approaches. From electronic structure methods suitable to describe transition metal complexes to approaches able to simulate their nuclear dynamics under the effect of light, we give particular attention to build a bridge between theory and experiment by critically discussing the different models commonly adopted in the interpretation of spectroscopic experiments and the simulation of particular observables. Thereby, we review all the studies of excited-state dynamics on transition metal complexes, both in gas phase and in solution from reduced to full dimensionality.

Validating fewest-switches surface hopping in the presence of laser fields

The capability of fewest-switches surface hopping (FSSH) to describe non-adiabatic dynamics under explicit excitation with external fields is evaluated. Different FSSH parameters are benchmarked against multi-configurational time dependent Hartree (MCTDH) reference calculations using SO₂ and 2-thiocytosine as model, yet realistic, molecular systems. Qualitatively, FSSH is able to reproduce the trends in the MCTDH dynamics with (also without) an explicit external field; however, no set of FSSH parameters is ideal. The adequate treatment of the overcoherence in FSSH is revealed as the driving factor to improve the description of the excitation process with respect to the MCTDH reference. Here, two corrections were tested: the augmented-FSSH (AFSSH) correction and the energy-based decoherence correction. A dependence on the employed basis is detected in AFSSH, performing better when spin-orbit and external laser field couplings are treated as off-diagonal elements instead of projecting them onto the diagonal of the Hamilton operator. In the presence of an electric field, the excited state dynamics was found to depend strongly on the vector used to rescale the kinetic energy along after a transition between surfaces. For SO₂, recurrence of the excited wave packet throughout the duration of the applied laser pulse is observed for laser pulses (>100 fs), resulting in additional interferences missed by FSSH and only visible in variational multi-configurational Gaussian when utilizing a large number of Gaussian basis functions. This feature vanishes when going toward larger molecules, such as 2-thiocytosine, where this effect is barely visible in a laser pulse 200 fs long.

Analysis of bath motion in MM-SQC dynamics via dimensionality reduction approach: Principal component analysis

The system-plus-bath model is an important tool to understand the nonadiabatic dynamics of large molecular systems. Understanding the collective motion of a large number of bath modes is essential for revealing their key roles in the overall dynamics. Here, we applied principal component analysis (PCA) to investigate the bath motion in the basis of a large dataset generated from the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian nonadiabatic dynamics for the excited-state energy transfer in the Frenkel-exciton model. The PCA method clearly elucidated that two types of bath modes, which either display strong vibronic coupling or have frequencies close to that of the electronic transition, are important to the nonadiabatic dynamics. These observations were fully consistent with the physical insights. The conclusions were based on the PCA of the trajectory data and did not involve significant pre-defined physical knowledge. The results show that the PCA approach, which is one of the simplest unsupervised machine learning dimensionality reduction methods, is a powerful one for analyzing complicated nonadiabatic dynamics in the condensed phase with many degrees of freedom.

Behind the scenes of spin-forbidden decay pathways in transition metal complexes

The interpretation of the ultrafast photophysics of transition metal complexes following photo-absorption is quite involved as the heavy metal center leads to a complicated and entangled singlet-triplet manifold. This opens up multiple pathways for deactivation, often with competitive rates. As a result, intersystem crossing (ISC) and phosphorescence are commonly observed in transition metal complexes. A detailed understanding of such an excited-state structure and dynamics calls for state-of-the-art experimental and theoretical methodologies. In this review, we delve into the inability of non-relativistic quantum theory to describe spin-forbidden transitions, which can be overcome by taking into account spin-orbit coupling, whose importance grows with increasing atomic number. We present the quantum chemical theory of phosphorescence and ISC together with illustrative examples. Finally, a few applications are highlighted, bridging the gap between theoretical studies and experimental applications, such as photofunctional materials.

Photoinduced Dynamics with Constrained Vibrational Motion: FrozeNM Algorithm

Ab initio molecular dynamics (AIMD) simulation, analyzed in terms of vibrational normal modes, is a widely used technique that facilitates understanding of complex structural motions and coupling between electronic and nuclear degrees of freedom. Usually, only a subset of vibrations is directly involved in the process of interest. The impact of these vibrations can be evaluated by performing AIMD simulations by selectively freezing certain motions. Herein, we present frozen normal mode (FrozeNM), a new algorithm to apply normal-mode constraints in AIMD simulations, as implemented in the nonadiabatic excited state molecular dynamics code. We further illustrate its capacity by analyzing the impact of normal-mode constraints on the photoinduced energy transfer between polyphenylene ethynylene dendrimer building blocks. Our results show that the electronic relaxation can be significantly slowed down by freezing a well-selected small subset of active normal modes characterized by their contributions in the direction of energy transfer. The application of these constraints reduces the nonadiabatic coupling between electronic excited states during the entire dynamical simulations. Furthermore, we validate reduced dimensionality models by freezing all the vibrations, except a few active modes. Altogether, we consider FrozeNM as a useful tool that can be broadly used to underpin the role of vibrational motion in a studied process and to formulate reduced models that describe essential physical phenomena.

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

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