Spin-vibronic quantum dynamics for ultrafast excited-state processes

Simulation of Ultrafast Excited-State Dynamics in Fe(II) Complexes: Assessment of Electronic Structure Descriptions

The assessment of electronic structure descriptions utilized in the simulation of the ultrafast excited-state dynamics of Fe(II) complexes is presented. Herein, we evaluate the performance of the RPBE, OPBE, BLYP, B3LYP, B3LYP*, PBE0, TPSSh, CAM-B3LYP, and LC-BLYP (time-dependent) density functional theory (DFT/TD-DFT) methods in full-dimensional trajectory surface hopping (TSH) simulations carried out on linear vibronic coupling (LVC) potentials. We exploit the existence of time-resolved X-ray emission spectroscopy (XES) data for the [Fe(bmip)2]2+ and [Fe(terpy)2]2+ prototypes for dynamics between metal-to-ligand charge-transfer (MLCT) and metal-centered (MC) states, which serve as a reference to benchmark the calculations (bmip = 2,6-bis(3-methyl-imidazole-1-ylidine)-pyridine, terpy = 2,2':6',2″-terpyridine). The results show that the simulated ultrafast population dynamics between MLCT and MC states with various spin multiplicities (singlet, triplet, and quintet) highly depend on the utilized DFT/TD-DFT method, with the percentage of exact (Hartree-Fock) exchange being the governing factor. Importantly, B3LYP* and TPSSh are the only DFT/TD-DFT methods with satisfactory performance, best reproducing the experimentally resolved dynamics for both complexes, signaling an optimal balance in the description of MLCT-MC energetics. This work demonstrates the power of combining TSH/LVC dynamics simulations with time-resolved experimental reference data to benchmark full-dimensional potential energy surfaces.

Quantification of memory effects in topological two-band open quantum systems

We incorporate non-Markovian profiles and Linear Response Theory to analyze memory effects in two-band topological quantum systems. Furthermore, we have applied a measure of non-Markovianity in terms of nonlinear optical spectroscopy. On the other hand, we resort to memory kernel, solve the integro-differential equation of the open two-band topological quantum system to describe the degrees of non-Markovianity, calculate response factors based on Linear Response Theory, and analyze non-Markovian dynamics by varying the parameters of the nonlinear spectroscopy environment of the respective open quantum system.

Tuning ultrafast time-evolution of photo-induced charge-transfer states: A real-time electronic dynamics study in substituted indenotetracene derivatives

Photo-induced charge transfer (CT) states are pivotal in many technological and biological processes. A deeper knowledge of such states is mandatory for modeling the charge migration dynamics. Real-time time-dependent density functional theory (RT-TD-DFT) electronic dynamics simulations are employed to explicitly observe the electronic density time-evolution upon photo-excitation. Asymmetrically substituted indenotetracene molecules, given their potential application as n-type semiconductors in organic photovoltaic materials, are here investigated. Effects of substituents with different electron-donating characters are analyzed in terms of the overall electronic energy spacing and resulting ultrafast CT dynamics through linear response (LR-)TD-DFT and RT-TD-DFT based approaches. The combination of the computational techniques here employed provided direct access to the electronic density reorganization in time and to its spatial and rational representation in terms of molecular orbital occupation time evolution. Such results can be exploited to design peculiar directional charge dynamics, crucial when photoactive materials are used for light-harvesting applications.

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

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

Watching the Interplay between Photoinduced Ultrafast Charge Dynamics and Nuclear Vibrations

Here is presented the ultrafast hole-electron dynamics of photoinduced metal to ligand charge-transfer (MLCT) states in a Ru(II) complex, [Ru(dcbpy)2(NCS)2]4- (dcbpy = 4,4'-dicarboxy-2,2'-bipyridine), a photoactive molecule employed in dye sensitized solar cells. Via cutting-edge computational techniques, a tailored computational protocol is here presented and developed to provide a detailed analysis of the electronic manifold coupled with nuclear vibrations to better understand the nonradiative pathways and the resulting overall dye performances in light-harvesting processes (electron injection). Thus, the effects of different vibrational modes were investigated on both the electronic levels and charge transfer dynamics through a theoretical-computational approach. First, the linear response time-dependent density functional (LR-TDDFT) formalism was employed to characterize excitation energies and spacing among electronic levels (the electronic layouts). Then, to understand the ultrafast (femtosecond) charge dynamics on the molecular scale, we relied on the nonperturbative mean-field quantum electronic dynamics via real-time (RT-) TDDFT. Three vibrational modes were selected, representative for collective nuclear movements that can have a significant influence on the electronic structure: two involving NCS- ligands and one involving dcbpy ligands. As main results, we observed that such MLCT states, under vibrational distortions, are strongly affected and a faster interligand electron transfer mechanism is observed along with an increasing MLCT character of the adiabatic electronic states approaching closer in energy due to the vibrations. Such findings can help both in providing a molecular picture of multidimensional vibro-electronic spectroscopic techniques, used to characterize ultrafast coherent and noncoherent dynamics of complex systems, and to improve dye performances with particular attention to the study of energy or charge transport processes and vibronic couplings.

Nonadiabatic Dynamics Simulations for Photoinduced Processes in Molecules and Semiconductors: Methodologies and Applications

Nonadiabatic dynamics (NAMD) simulations have become powerful tools for elucidating complicated photoinduced processes in various systems from molecules to semiconductor materials. In this review, we present an overview of our recent research on photophysics of molecular systems and periodic semiconductor materials with the aid of ab initio NAMD simulation methods implemented in the generalized trajectory surface-hopping (GTSH) package. Both theoretical backgrounds and applications of the developed NAMD methods are presented in detail. For molecular systems, the linear-response time-dependent density functional theory (LR-TDDFT) method is primarily used to model electronic structures in NAMD simulations owing to its balanced efficiency and accuracy. Moreover, the efficient algorithms for calculating nonadiabatic coupling terms (NACTs) and spin-orbit couplings (SOCs) have been coded into the package to increase the simulation efficiency. In combination with various analysis techniques, we can explore the mechanistic details of the photoinduced dynamics of a range of molecular systems, including charge separation and energy transfer processes in organic donor-acceptor structures, ultrafast intersystem crossing (ISC) processes in transition metal complexes (TMCs), and exciton dynamics in molecular aggregates. For semiconductor materials, we developed the NAMD methods for simulating the photoinduced carrier dynamics within the framework of the Kohn-Sham density functional theory (KS-DFT), in which SOC effects are explicitly accounted for using the two-component, noncollinear DFT method. Using this method, we have investigated the photoinduced carrier dynamics at the interface of a variety of van der Waals (vdW) heterojunctions, such as two-dimensional transition metal dichalcogenides (TMDs), carbon nanotubes (CNTs), and perovskites-related systems. Recently, we extended the LR-TDDFT-based NAMD method for semiconductor materials, allowing us to study the excitonic effects in the photoinduced energy transfer process. These results demonstrate that the NAMD simulations are powerful tools for exploring the photodynamics of molecular systems and semiconductor materials. In future studies, the NAMD simulation methods can be employed to elucidate experimental phenomena and reveal microscopic details as well as rationally design novel photofunctional materials with desired properties.

Understanding Charge Dynamics in Dense Electronic Manifolds in Complex Environments

Photoinduced charge transfer (CT) excited states and their relaxation mechanisms can be highly interdependent on the environment effects and the consequent changes in the electronic density. Providing a molecular interpretation of the ultrafast (subpicosecond) interplay between initial photoexcited states in such dense electronic manifolds in condensed phase is crucial for improving and understanding such phenomena. Real-time time-dependent density functional theory is here the method of choice to observe the charge density, explicitly propagated in an ultrafast time domain, along with all time-dependent properties that can be easily extracted from it. A designed protocol of analysis for real-time electronic dynamics to be applied to time evolving electronic density related properties to characterize both in time and in space CT dynamics in complex systems is here introduced and validated, proposing easy to be read cross-correlation maps. As case studies to test such tools, we present the photoinduced charge-transfer electronic dynamics of 5-benzyluracil, a mimic of nucleic acid/protein interactions, and the metal-to-ligand charge-transfer electronic dynamics in water solution of [Ru(dcbpy)2(NCS)2]4-, dcbpy = (4,4'-dicarboxy-2,2'-bipyridine), or "N34-", a dye sensitizer for solar cells. Electrostatic and explicit ab initio treatment of solvent molecules have been compared in the latter case, revealing the importance of the accurate modeling of mutual solute-solvent polarization on CT kinetics. We observed that explicit quantum mechanical treatment of solvent slowed down the charge carriers mobilities with respect to the gas-phase. When all water molecules were modeled instead as simpler embedded point charges, the electronic dynamics appeared enhanced, with a reduced hole-electron distance and higher mean velocities due to the close fixed charges and an artificially increased polarization effect. Such analysis tools and the presented case studies can help to unveil the influence of the electronic manifold, as well as of the finite temperature-induced structural distortions and the environment on the ultrafast charge motions.

High Intrinsic Phosphorescence Efficiency and Density Functional Theory Modeling of Ru(II)-Bipyridine Complexes with π-Aromatic-Rich Cyclometalated Ligands: Attributions of Spin-Orbit Coupling Perturbation and Efficient Configurational Mixing of Singlet Excited States

A series of π-aromatic-rich cyclometalated ruthenium(II)-(2,2'-bipyridine) complexes ([Ru(bpy)2(πAr-CM)]+) in which πAr-CM is diphenylpyrazine or 1-phenylisoquinoline were prepared. The [Ru(bpy)2(πAr-CM)]+ complexes had remarkably high phosphorescence rate constants, k RAD(p), and the intrinsic phosphorescence efficiencies (ιem(p) = k RAD(p)/(νem(p))3) of these complexes were found to be twice the magnitudes of simply constructed cyclometalated ruthenium(II) complexes ([Ru(bpy)2(sc-CM)]+), where νem(p) is the phosphorescence frequency and sc-CM is 2-phenylpyridine, benzo[h]quinoline, or 2-phenylpyrimidine. Density functional theory (DFT) modeling of the [Ru(bpy)2(CM)]+ complexes indicated numerous singlet metal-to-ligand charge transfers for 1MLCT-(Ru-bpy) and 1MLCT-(Ru-CM), excited states in the low-energy absorption band and 1ππ*-(aromatic ligand) (1ππ*-LAr) excited states in the high-energy band. DFT modeling of these complexes also indicated phosphorescence-emitting state (Te) configurations with primary MLCT-(Ru-bpy) characteristics. The variation in ιem(p) for the spin-forbidden Te (3MLCT-(Ru-bpy)) excited state of the complex system that was examined in this study can be understood through the spin-orbit coupling (SOC)-mediated sum of intensity stealing (∑SOCM-IS) contribution from the primary intensity of the low-energy 1MLCT states and second-order intensity perturbation from the significant configuration between the low-energy 1MLCT and high-energy intense 1ππ*-LAr states. In addition, the observation of unusually high ιem(p) magnitudes for these [Ru(bpy)2(πAr-CM)]+ complexes can be attributed to the values for both intensity factors in the ∑SOCM-IS formalism being individually greater than those for [Ru(bpy)2(sc-CM)]+ ions.

Nonadiabatic excited-state dynamics of ReCl(CO)3(bpy) in two different solvents

We present a study of excited-states relaxation of the complex ReCl(CO)₃(bpy) (bpy = 2,2-bipyridine) using a nonadiabatic TD-DFT dynamics on spin-mixed potential energy surfaces in explicit acetonitrile (ACN) and dimethylsulfoxide (DMSO) solutions up to 800 fs. ReCl(CO)₃(bpy) belongs to a group of important photosensitizers which show ultrafast biexponential subpicosecond fluorescence decay kinetics. The choice of solvents was motivated by the different excited-state relaxation dynamics observed in subpicosecond time-resolved IR (TRIR) experiments. Simulations of intersystem crossing (ISC) showed the development of spin-mixed states in both solvents. Transformation of time-dependent populations of spin-mixed states enabled to monitor the temporal evolution of individual singlet and triplet states, fitting of bi-exponential decay kinetics, and simulating the time-resolved fluorescence spectra that show only minor differences between the two solvents. Analysis of structural relaxation and solvent reorganization employing time-resolved proximal distribution functions pointed to the factors influencing the fluorescence decay time constants. Nonadiabatic dynamics simulations of time-evolution of electronic, molecular, and solvent structures emerge as a powerful technique to interpret time-resolved spectroscopic data and ultrafast photochemical reactivity.

Spin–vibronic coupling in the quantum dynamics of a Fe(III) trigonal-bipyramidal complex

The presence of a high density of excited electronic states in the immediate vicinity of the optically bright state of a molecule paves the way for numerous photo-relaxation channels. In transition-metal complexes, the presence of heavy atoms results in a stronger spin–orbit coupling, which enables spin forbidden spin-crossover processes to compete with the spin-allowed internal conversion processes. However, no matter how effectively the states cross around the Franck–Condon region, the degree of vibronic coupling, of both relativistic and non-relativistic nature, drives the population distribution among these states. One such case is demonstrated in this work for the intermediate-spin Fe(III) trigonal-bipyramidal complex. A quantum dynamical investigation of the photo-deactivation mechanism in the Fe(III) system is presented using the multi-configurational time-dependent Hartree approach based on the vibronic Hamiltonian whose coupling terms are derived from the state-averaged complete active space self-consistent field/complete active space with second-order perturbation theory (CASPT2) calculations and spin–orbit coupling of the scalar-relativistic CASPT2 states. The results of this study show that the presence of a strong (non-relativistic) vibronic coupling between the optically bright intermediate-spin state and other low-lying states of the same spin-multiplicity overpowers the spin–orbit coupling between the intermediate-spin and high-spin states, thereby lowering the chances of spin-crossover while exhibiting ultrafast relaxation among the intermediate-spin states. In a special case, where the population transfer pathway via the non-relativistic vibronic coupling is blocked, the probability of the spin-crossover is found to increase. This suggests that a careful modification of the complex by incorporation of heavier atoms with stronger relativistic effects can enhance the spin-crossover potential of Fe(III) intermediate-spin complexes.

Rationalizing the Fluorescence Behavior of Core-Substituted Naphthalene Diimides

We study the internal conversion (IC) and intersystem crossing (ISC) pathways of low-lying excited electronic states of three core-substituted naphthalene diimides (bNDI, yNDI, and gNDI) using wavepacket simulations within the linear vibronic coupling method. Our wavepacket simulations reproduce the experimental electronic absorption spectra very well. All molecules decay rapidly to S₂ upon populating a higher dipole-allowed singlet excited-state. The S₂ → S₁ IC dynamics and singlet-triplet energy gap, spin-orbit coupling strength trends suggest a favorable S₂ → T₄ ISC in gNDI. The efficient ultrafast T₄ formation and its decay to lower triplet states make gNDI nonfluorescent. Such triplet formation pathways are not operative in both bNDI and yNDI; hence, these molecules emit fluorescence from S₁ after a slower S₂ → S₁ IC.

Toward Simulation of Fe(II) Low-Spin → High-Spin Photoswitching by Synergistic Spin-Vibronic Dynamics

A new theoretical approach is presented and applied for the simulation of Fe(II) low-spin (LS, singlet, t_2g⁶e_g⁰) → high-spin (HS, quintet, t_2g⁴e_g²) photoswitching dynamics of the octahedral model complex [Fe(NCH)₆]²⁺. The utilized synergistic methodology heavily exploits the strengths of complementary electronic structure and spin-vibronic dynamics methods. Specifically, we perform 3D quantum dynamics (QD) and full-dimensional trajectory surface hopping (TSH, in conjunction with a linear vibronic coupling model), with the modes for QD selected by TSH. We follow a hybrid approach which is based on the application of time-dependent density functional theory (TD-DFT) excited-state potential energy surfaces (PESs) and multiconfigurational second-order perturbation theory (CASPT2) spin–orbit couplings (SOCs). Our method delivers accurate singlet–triplet–quintet intersystem crossing (ISC) dynamics, as assessed by comparison to our recent high-level ab initio simulations and related time-resolved experimental data. Furthermore, we investigate the capability of our simulations to identify the location of ISCs. Finally, we assess the approximation of constant SOCs (calculated at the Franck–Condon geometry), whose validity has central importance for the combination of TD-DFT PESs and CASPT2 SOCs. This efficient methodology will have a key role in simulating LS → HS dynamics for more complicated cases, involving higher density of states and varying electronic character, as well as the analysis of ultrafast experiments.

Intersystem crossing and internal conversion dynamics with GAIMS-TeraChem: Excited state relaxation in 2-cyclopentenone

Excited states relaxation in complex molecules often involves two types of nonradiative transitions, internal conversion (IC) and intersystem crossing (ISC). In the situations when the timescales of IC and ISC are comparable, an interplay between these two types of transitions can lead to complex nonadiabatic dynamics on multiple electronic states of different characters and spin multiplicities. We demonstrate that the generalized ab initio multiple spawning (GAIMS) method interfaced with the fast graphics processing unit-based TeraChem electronic structure code can be used to model such nonadiabatic dynamics involving both the IC and ISC transitions in molecules of moderate size. We carried out 1500 fs GAIMS simulations leading to the creation of up to 2500 trajectory basis functions to study the excited states relaxation in 2-cyclopentenone. After a vertical excitation from the ground state to the bright S₂ state, the molecule quickly relaxes to the S₁ state via conical intersection. The following relaxation proceeds along two competing pathways: one involves IC to the ground state, and the other is dominated by ISC to the low-lying triplet states. The time constants describing the population transfer between the six lowest singlet and triplet states predicted by the GAIMS dynamics are in good agreement with the characteristic times of IC and ISC obtained from the analysis of the time-resolved photoelectron spectrum.

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.

Excited-state dynamics of [Mn(im)(CO)3(phen)]+: PhotoCORM, catalyst, luminescent probe?

Mn(I) α-diimine carbonyl complexes have shown promise in the development of luminescent CO release materials (photoCORMs) for diagnostic and medical applications due to their ability to balance the energy of the low-lying metal-to-ligand charge transfer (MLCT) and metal-centered (MC) states. In this work, the excited state dynamics of [Mn(im)(CO)₃(phen)]⁺ (im = imidazole; phen = 1,10-phenanthroline) is investigated by means of wavepacket propagation on the potential energy surfaces associated with the 11 low-lying S_n singlet excited states within a vibronic coupling model in a (quasi)-diabatic representation including 16 nuclear degrees of freedom. The results show that the early time photophysics (<400 fs) is controlled by the interaction between two MC dissociative states, namely, S₅ and S₁₁, with the lowest S₁-S₃ MLCT bound states. In particular, the presence of S₁/S₅ and S₂/S₁₁ crossings within the diabatic picture along the Mn-CO_axial dissociative coordinate (q_Mn-COaxial) favors a two-stepwise population of the dissociative states, at about 60-70 fs (S₁₁) and 160-180 fs (S₅), which reaches about 10% within 200 fs. The one-dimensional reduced densities associated with the dissociative states along q_Mn-COaxial as a function of time clearly point to concurrent primary processes, namely, CO release vs entrapping into the S₁ and S₂ potential wells of the lowest luminescent MLCT states within 400 fs, characteristics of luminescent photoCORM.

Modeling Spin-Crossover Dynamics

In this article, we review nonadiabatic molecular dynamics (NAMD) methods for modeling spin-crossover transitions. First, we discuss different representations of electronic states employed in the grid-based and direct NAMD simulations. The nature of interstate couplings in different representations is highlighted, with the main focus on nonadiabatic and spin-orbit couplings. Second, we describe three NAMD methods that have been used to simulate spin-crossover dynamics, including trajectory surface hopping, ab initio multiple spawning, and multiconfiguration time-dependent Hartree. Some aspects of employing different electronic structure methods to obtain information about potential energy surfaces and interstate couplings for NAMD simulations are also discussed. Third, representative applications of NAMD to spin crossovers in molecular systems of different sizes and complexities are highlighted. Finally, we pose several fundamental questions related to spin-dependent processes. These questions should be possible to address with future methodological developments in NAMD.

Quantum and semiclassical dynamical studies of nonadiabatic processes in solution: achievements and perspectives

We concisely review the main methodological approaches to model nonadiabatic dynamics in isotropic solutions and their applications. Three general classes of models are identified as the most used to include solvent effects in the simulations. The first model describes the solvent as a set of harmonic collective modes coupled to the solute degrees of freedom, and the second as a continuum, while the third explicitly includes solvent molecules in the calculations. The issues related to the use of these models in semiclassical and quantum dynamical simulations are discussed, as well as the main limitations and perspectives of each approach.

Parameterization of a linear vibronic coupling model with multiconfigurational electronic structure methods to study the quantum dynamics of photoexcited pyrene

With this work, we present a protocol for the parameterization of a Linear Vibronic Coupling (LVC) Hamiltonian for quantum dynamics using highly accurate multiconfigurational electronic structure methods such as RASPT2/RASSCF, combined with a maximum-overlap diabatization technique. Our approach is fully portable and can be applied to many medium-size rigid molecules whose excited state dynamics requires a quantum description. We present our model and discuss the details of the electronic structure calculations needed for the parameterization, analyzing critical situations that could arise in the case of strongly interacting excited states. The protocol was applied to the simulation of the excited state dynamics of the pyrene molecule, starting from either the first or the second bright state (S₂ or S₅). The LVC model was benchmarked against state-of-the-art quantum mechanical calculations with optimizations and energy scans and turned out to be very accurate. The dynamics simulations, performed including all active normal coordinates with the multilayer multiconfigurational time-dependent Hartree method, show good agreement with the available experimental data, endorsing prediction of the excited state mechanism, especially for S₅, whose ultrafast deactivation mechanism was not yet clearly understood.

Near-IR Charge-Transfer Emission at 77 K and Density Functional Theory Modeling of Ruthenium(II)-Dipyrrinato Chromophores: High Phosphorescence Efficiency of the Emitting State Related to Spin-Orbit Coupling Mediation of Intensity from Numerous Low-Energy Singlet Excited States

Efficient charge-transfer (CT) phosphorescence in the near-IR (NIR) spectral region is reported for four substituted Ru-(R-dipyrrinato) complexes, [Ru(bpy)₂(R-dipy)](PF₆), where bpy is 2,2'-bipyridine and the substituent R is phenyl (ph), 2,4,6-trimethylphenyl, 4-carboxyphenyl (HOOC-ph), or 4-pyridinyl. The experimentally determined phosphorescence efficiency, ι_em(p) = k_RAD(p)/(ν_em(p))³ (where k_RAD(p) and ν_em(p) are the phosphorescence rate constant and the phosphorescence frequency, respectively), of the [Ru(bpy)₂(R-dipy)]⁺ complexes was approximately double that of [Ru(bpy)(Am)₄]²⁺ complexes (Am = ammine ligand) in the NIR region. Density functional theory (DFT) modeling indicated two strikingly different electronic configurations of the triplet emitting state (T_e) in the two types of complexes. The T_e of [Ru(bpy)₂(R-dipy)]⁺ complexes shows a CT-type corresponding to the metal-to-ligand charge transfer (MLCT)-(Ru-(R-dipy)) and the ππ*-(R-dipy) moiety configurations, and the T_e state in the [Ru(bpy)(Am)₄]²⁺ complexes corresponds to an approximately MLCT excited state consisting of mostly MLCT-(Ru-bpy) with a minimal ππ*(bpy) contribution. DFT modeling also indicated that the low-energy singlet excited states in the T_e geometry (S_n(T)) of the [Ru(bpy)₂(ph-dipy)]⁺ complex consist of numerous CT-S_n(T)-type states of the Ru-dipy and Ru-bpy moieties, whereas the [Ru(bpy)(Am)₄]²⁺ ions show quite simple MLCT-S_n(T)-type states of the Ru-bpy moiety. Based on experimental observations, DFT modeling, and the plain spin-orbit coupling (SOC) principle, we conclude that the remarkably high ι_em(p) amplitudes of the [Ru(bpy)₂(R-dipy)]⁺ complexes relative to those of [Ru(bpy)(Am)₄]²⁺ complexes can be attributed to the relatively substantial contribution of intrinsic SOC-mediated intensity stealing from the numerous low-energy CT-type S_n(T) states.

Ultrafast processes: coordination chemistry and quantum theory

Coordination compounds, characterized by fascinating and tunable electronic properties, are capable of binding easily to proteins, polymers, wires and DNA. Upon irradiation, these molecular systems develop functions finding applications in solar cells, photocatalysis, luminescent and conformational probes, electron transfer triggers and diagnostic or therapeutic tools. The control of these functions is activated by the light wavelength, the metal/ligand cooperation and the environment within the first picoseconds (ps). After a brief summary of the theoretical background, this perspective reviews case studies, from 1st row to 3rd row transition metal complexes, that illustrate how spin-orbit, vibronic coupling and quantum effects drive the photophysics of this class of molecules at the early stage of the photoinduced elementary processes within the fs-ps time scale range.

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.

Formation of excited triplet states in naphthalene diimide and perylene diimide derivatives: A detailed theoretical study

Mechanistic details of the excited triplet state formation upon photoexcitation to the low-lying singlet manifold in naphthalene diimide and perylene diimide derivatives are explored theoretically. Static and dynamic aspects of two singlets (S₁ and S₂) and six triplets (T₁-T₆) of these molecules are investigated. Suitable vibronic Hamiltonians are constructed to investigate the internal conversion dynamics in both the singlet and triplet manifolds. Computed singlet-triplet energetics, spin-orbit coupling matrix elements, and intersystem crossing rates strongly suggest an efficient intersystem crossing process involving higher triplet states (T₆, T₅, and T₄). Separate full dimensional quantum wavepacket simulations of singlet and triplet manifolds in the approximate linear vibronic model by assuming initial Franck-Condon conditions are carried out to unravel the internal conversion decay dynamics in the respective manifolds. The obtained diabatic electronic populations and nuclear densities are analyzed to illustrate the triplet generation pathways involving higher triplet states in these molecules.

Spin-inversion mechanisms in O2 binding to a model heme compound: A perspective from nonadiabatic wave packet calculations

Spin-inversion dynamics in O₂ binding to a model heme complex, which consisted of Fe(II)-porphyrin and imidazole, were studied using nonadiabatic wave packet dynamics calculations. We considered three active nuclear degrees of freedom in the dynamics, including the motions along the Fe-O distance, Fe-O-O angle, and Fe out-of-plane distance. Spin-free potential energy surfaces for the singlet, triplet, quintet, and septet states were developed using density functional theory calculations, and spin-orbit coupling elements were obtained from CASSCF-level electronic structure calculations. The spin-inversion mainly occurred between the singlet state and one of the triplet states due to large spin-orbit couplings and the contributions of other states were extremely small. The present quantum dynamics calculations suggested that the narrow crossing region model plays a dominant role in the O₂ binding dynamics. In addition, the one-dimensional Landau-Zener model underestimated the nonadiabatic transition probability.

Time-Resolved Femtosecond Stimulated Raman Spectra and DFT Anharmonic Vibrational Analysis of an Electronically Excited Rhenium Photosensitizer

Time-resolved femtosecond stimulated Raman spectra (FSRS) of a prototypical organometallic photosensitizer/photocatalyst ReCl(CO)₃(2,2'-bipyridine) were measured in a broad spectral range ∼40-2000 (4000) cm^-1 at time delays from 40 fs to 4 ns after 400 nm excitation of the lowest allowed electronic transition. Theoretical ground- and excited-state Raman spectra were obtained by anharmonic vibrational analysis using second-order vibrational perturbation theory on vibrations calculated by harmonic approximation at density functional theory-optimized structures. A good match with anharmonically calculated vibrational frequencies allowed for assigning experimental Raman features to particular vibrations. Observed frequency shifts upon excitation (ν(ReCl) and ν(CC inter-ring) vibrations upward; ν(CC, CN) and ν(Re-C) downward) are consistent with the bonding/antibonding characters of the highest occupied molecular orbital and the lowest unoccupied molecular orbital involved in excitation and support the delocalized formulation of the lowest triplet state as ReCl(CO)₃ → bpy charge transfer. FSRS spectra show a mode-specific temporal evolution, providing insights into the intersystem crossing (ISC) mechanism and subsequent relaxation. Most of the Raman features are present at ∼40 fs and exhibit small shifts and intensity changes with time. The 1450-1600 cm^-1 group of bands due to CC, CN, and CC(inter-ring) stretching vibrations undergoes extensive restructuring between 40 and ∼150 fs, followed by frequency upshifts and a biexponential (0.38, 21 ps) area growth, indicating progressing charge separation in the course of the formation and relaxation of the lowest triplet state. Early (40-150 fs) restructuring was also observed in the low-frequency range for ν(Re-Cl) and δ(Re-C-O) vibrations that are presumably activated by ISC. FSRS experimental innovations employed to measure low- and high-energy Raman features simultaneously are described and discussed in detail.

Dissipation and dynamics in ultrafast intersystem crossings

The effects of dynamics and dissipation on ultrafast intersystem crossings are studied for a dissipative two-level system coupled to a local vibronic mode. A method of amplitude damping of the wave packet is presented that accounts better for the position of the wave packet and avoids spurious transitions between potential wells. It is demonstrated that Fermi's golden rule, the typical semiquantitative approach to extract population transfer rates from potential landscapes, only holds under limited conditions. Generally, the effects of dynamics and dissipation lead to deviations from the expected exponential population transfer, strong changes in transfer times and total population transfer, and significant recurrence or "spill back" of the wave packet.

Competition between the heavy atom effect and vibronic coupling in donor–bridge–acceptor organometallics

The excited state properties and intersystem crossing dynamics of a series of donor–bridge–acceptor carbene metal-amides based upon the coinage metals Cu, Ag, Au, are investigated using quantum dynamics simulations and supported by photophysical characterisation.

Ultrafast dynamics in LMCT and intraconfigurational excited states in hexahaloiridates(iv), models for heavy transition metal complexes and building blocks of quantum correlated materials

Two heavy octahedral Ir(iv) halides in intraconfigurational and LMCT excited electronic states with ultrafast relaxation dynamics driven by the Jahn–Teller effect.

Unconventional two-step spin relaxation dynamics of [Re(CO)3(im)(phen)]+ in aqueous solution

Changes of molecular spin are ubiquitous in chemistry and biology. Among spin flip processes, one of the fastest is intersystem crossing (ISC) in transition metal complexes. Here, we investigate the spin relaxation dynamics and emission spectrum of [Re(CO)3(im)(phen)]+ (im = imidazole, phen = phenanthroline) using extensive full-dimensional excited-state dynamics simulations in explicit aqueous solution. Contrary to what has been observed in other transition metal complexes, the transition from the singlet to triplet states occurs via a two-step process, with clearly separable electronic and nuclear-driven components with two different time scales. The initially excited electronic wave function is a "molecular spin-orbit wave packet" that evolves almost instantaneously, with an 8 fs time constant, into an approximate 25 : 75 singlet-to-triplet equilibrium. Surprisingly, this ISC process is an order of magnitude faster than it was previously documented for this and other rhenium(i) carbonyl diimine complexes from emission spectra. Simulations including explicit laser field interactions evidence that few-cycle UV laser pulses are required to follow the creation and evolution of such molecular spin-orbit wave packets. The analysis of the dynamics also reveals a retarded ISC component, with a time constant of 420 fs, which can be explained invoking intramolecular vibrational energy redistribution. The emission spectrum is shown to be characterized by ISC convoluted with internal conversion and vibrational relaxation. These results provide fundamental understanding of ultrafast intersystem crossing in transition metal complexes.

Light-Induced Spin Crossover in an Intermediate-Spin Penta-Coordinated Iron(III) Complex

(PMe₃)₂FeCl₃ is an Fe(III) complex that exists in the intermediate-spin ground state in a distorted trigonal bipyramidal geometry. An electronic state with high-spin configuration lies in close vicinity to the ground state, making it a potential spin crossover candidate. A mechanistic account of the spin crossover from the lowest quartet state (Q₀) to the lowest sextet state (S₁) of this complex is provided by exploring both thermal and light-induced pathways. The presence of a large barrier between the two spin states suggests a possible thermal spin crossover at a rather high temperature. The light-induced spin crossover is investigated by employing complete active space self-consistent field calculations together with dynamic correlation and spin-orbit coupling for the lowest seven quartet and lowest five sextet states. The system in the Q₀ state upon light absorption is excited to the optically bright Q₄ LMCT state. By following minimum energy pathways along the electronic states, two light-induced pathways for spin crossover are identified. From the Q₄ state, the system can photo-regenerate the ground intermediate-spin state (Q₀) through an internal conversion of Q₄/Q₃ followed by Q₃/S₁ and S₁/Q₀ intersystem crossings. In an alternate route, through Q₄/S₂ intersystem crossing followed by S₂/S₁ internal conversion, the system can complete the spin crossover from the Q₀ to S₁ state.

Performance of trajectory surface hopping method in the treatment of ultrafast intersystem crossing dynamics

We carried out extensive studies to examine the performance of the fewest-switches surface hopping method in the description of the ultrafast intersystem crossing dynamic of various singlet-triplet (S-T) models by comparison with the results of the exact full quantum dynamics. Different implementation details and some derivative approaches were examined. As expected, it is better to perform the trajectory surface hopping calculations in the spin-adiabatic representation or by the local diabatization approach, instead of in the spin-diabatic representation. The surface hopping method provides reasonable results for the short-time dynamics in the S-T model with weak spin-orbital coupling (diabatic coupling), although it does not perform well in the models with strong spin-orbital coupling (diabatic coupling). When the system accesses the S-T potential energy crossing with rather high kinetic energy, the trajectory surface hopping method tends to produce a good description of the nonadiabatic intersystem crossing dynamics. The impact of the decoherence correction on the performance of the trajectory surface hopping is system dependent. It improves the result accuracy in many cases, while its influence may also be minor for other cases.

Charge-Transfer versus Charge-Separated Triplet Excited States of [ReI (dmp)(CO)3 (His124)(Trp122)]+ in Water and in Modified Pseudomonas aeruginosa Azurin Protein

A computational investigation of the triplet excited states of a rhenium complex electronically coupled with a tryptophan side chain and bound to an azurin protein is presented. In particular, by using high-level molecular modeling, evidence is provided for how the electronic properties of the excited-state manifolds strongly depend on coupling with the environment. Indeed, only upon explicitly taking into account the protein environment can two stable triplet states of metal-to-ligand charge transfer or charge-separated nature be recovered. In addition, it is also demonstrated how the rhenium complex plus tryptophan system in an aqueous environment experiences too much flexibility, which prevents the two chromophores from being electronically coupled. This occurrence disables the formation of a charge-separated state. The successful strategy requires a multiscale approach of combining molecular dynamics and quantum chemistry. In this context, the strategy used to parameterize the force fields for the electronic triplet states of the metal complex is also presented.

Understanding the potential for efficient triplet harvesting with hot excitons

Excited state energy transfer in disordered systems has attracted significant attention owing to the importance of this phenomenon in both artificial and natural systems that operate in electronically excited states. Of particular interest, especially in the context of organic electronics, is the dynamics of triplet excited states. Due to their weak coupling to the singlet manifold they can often act as low energy trapping sites and are therefore detrimental to device performance. Alternatively, by virtue of their long lifetime they can lead to enhanced diffusion lengths important for organic photovoltaics (OPV). Herein, we explore the triplet energy transfer mechanism from dichlorobenzene to thioxanthone in methanol solution. We rationalise previous experimental observations as arising from preferential population transfer into the lowest triplet state rather than the higher lying triplet state that is closer in energy. The reason for this is a delicate balance between the electronic coupling, reorganisation energy and the energy gap involved. The present results provide the understanding to potentially develop a hot exciton mechanism in materials for organic light emitting diodes (OLED) to achieve higher device efficiencies.

Highly efficient surface hopping dynamics using a linear vibronic coupling model

We report an implementation of the linear vibronic coupling (LVC) model within the surface hopping dynamics approach and present utilities for parameterizing this model in a blackbox fashion. This results in an extremely efficient method to obtain qualitative and even semi-quantitative information about the photodynamical behavior of a molecule, and provides a new route toward benchmarking the results of surface hopping computations. The merits and applicability of the method are demonstrated in a number of applications. First, the method is applied to the SO2 molecule showing that it is possible to compute its absorption spectrum beyond the Condon approximation, and that all the main features and timescales of previous on-the-fly dynamics simulations of intersystem crossing are reproduced while reducing the computational effort by three orders of magnitude. The dynamics results are benchmarked against exact wavepacket propagations on the same LVC potentials and against a variation of the electronic structure level. Four additional test cases are presented to exemplify the broader applicability of the model. The photodynamics of the isomeric adenine and 2-aminopurine molecules are studied and it is shown that the LVC model correctly predicts ultrafast decay in the former and an extended excited-state lifetime in the latter. Futhermore, the method correctly predicts ultrafast intersystem crossing in the modified nucleobase 2-thiocytosine and its absence in 5-azacytosine while it fails to describe the ultrafast internal conversion to the ground state in the latter.

Ultrafast carbon monoxide photolysis and heme spin-crossover in myoglobin via nonadiabatic quantum dynamics

Light absorption of myoglobin triggers diatomic ligand photolysis and a spin crossover transition of iron(II) that initiate protein conformational change. The photolysis and spin crossover reactions happen concurrently on a femtosecond timescale. The microscopic origin of these reactions remains controversial. Here, we apply quantum wavepacket dynamics to elucidate the ultrafast photochemical mechanism for a heme-carbon monoxide (heme-CO) complex. We observe coherent oscillations of the Fe-CO bond distance with a period of 42 fs and an amplitude of ∼1 Å. These nuclear motions induce pronounced geometric reorganization, which makes the CO dissociation irreversible. The reaction is initially dominated by symmetry breaking vibrations inducing an electron transfer from porphyrin to iron. Subsequently, the wavepacket relaxes to the triplet manifold in ∼75 fs and to the quintet manifold in ∼430 fs. Our results highlight the central role of nuclear vibrations at the origin of the ultrafast photodynamics of organometallic complexes.

Ultrafast Intersystem Crossing vs Internal Conversion in α-Diimine Transition Metal Complexes: Quantum Evidence

Whereas third row transition metal carbonyl α-diimine complexes display luminescent properties and possess low-lying triplet metal-to-ligand charge transfer (MLCT) states efficiently accessible by a spin-vibronic mechanism, first row analogues hold low-lying metal-centered (MC) excited states that could quench these properties. Upon visible irradiation, different functions are potentially stimulated, namely, luminescence, electron transfer, or photoinduced CO release, the branching ratio of which is governed by the energetics, the character, and the early time dynamics of the photoactive excited states. Simulations of ultrafast nonadiabatic quantum dynamics, including spin-vibronic effects, of [M(imidazole)(CO)₃(phenanthroline)]⁺ (M = Mn, Re) highlight the role of the metal atom. An ultrafast intersystem crossing process, driven by spin-orbit coupling, populates the low-lying triplet states of [Re(imidazole)(CO)₃(phen)]⁺ within the first tens of fs. In contrast, efficient internal conversion between the two lowest ¹MLCT states of [Mn(imidazole)(CO)₃(phen)]⁺ is mediated within 50 fs by vibronic coupling with upper MC and MLCT states.