Surface Hopping Method Research Articles

Modeling Charge Transport in Organic Semiconductors Using Neural Network Based Hamiltonians and Forces.

The fewest switches surface hopping method has been widely used for the simulation of charge transport in organic semiconductors. In the present study, we perform nonadiabatic molecular dynamics (NAMD) simulations of hole transport in anthracene and pentacene. The simulations employ neural network (NN) based Hamiltonians in two different nuclear relaxation schemes, which utilize either a precalculated reorganization energy or site energy gradients additionally obtained from NN models. The performance of the NN models is evaluated in reproducing hole mobilities and inverse participation ratios in terms of both quality and computational cost. The results show that charge mobilities and inverse participation ratios obtained by models, which were trained on DFTB or DFT training data, are in very good agreement with the respective QM reference method for implicit relaxation and, where available, also for explicit relaxation. Reasonable agreement with experimental hole mobilities is achieved. Utilizing our models in NAMD simulations of charge transfer amounts to a reduction of the computational cost in a range of 1 to 7 orders of magnitude compared to DFTB and DFT. This proves neural networks as promising tools for the improvement of accuracy and efficiency of charge and potentially also exciton transport simulations in complex and large molecular systems.

Large-scale surface hopping simulation of charge transport in hexagonal molecular crystals: role of electronic coupling signs

In large-scale surface hopping simulations with a huge number of electronic states, trivial crossings could easily lead to incorrect long-range charge transfer and induce large numerical errors. We here study the charge transport in two-dimensional hexagonal molecular crystals with a parameter-free full crossing corrected global flux surface hopping method. Fast time-step size convergence and system size independence have been realized in large systems containing thousands of molecular sites. In hexagonal systems, each molecular site has six nearest neighbours. We find that the signs of their electronic couplings have a strong impact on the charge mobility and delocalization strength. In particular, changing the signs of electronic couplings can even lead to a transition from hopping to band-like transport. In comparison, such phenomena cannot be observed in extensively studied two-dimensional square systems. This is attributed to symmetry of the electronic Hamiltonian and distribution of the energy levels. Due to its high performance, the proposed approach is promising to be applied to more realistic and complex systems for molecular design.

Mechanistic Insight and Intersystem Crossing Dynamics of the C(3P) + H2CO/D2CO Reaction

The reaction of atomic carbon, C(3P), with H2CO has been investigated using the direct dynamics trajectory surface hopping (DDTSH) method with Tully's fewest switches algorithm. The lowest lying ground triplet and single states are considered for the dynamics study at a reagent collision energy of 8.0 kcal/mol. From the trajectory calculations, we observed that CH2 + CO and H + HCCO are the two major product channels for the title reaction. The insertion mechanism of the C(3P) + H2CO reaction is rather complex and is followed by three distinct intermediates with no entrance channel barrier to the reaction on the B3LYP/6-31G(d,p) potential energy surfaces. The triplet insertion complexes are formed by three different approaches; "Sideways", "End-on" and "Head-on" attack of the triplet carbon atom toward H2CO molecule. Our dynamics calculations predict a new product channel (H + HCCO(X 2A'')) with a contribution of ∼46% of the overall products formation via ketocarbene intermediate through "Head-on" approach. Despite the weak spin-orbit coupling (SOC) interactions, intersystem crossing (ISC) via a ketocarbene intermediate has a small but significant contribution, about 2.3%, for the CH2 + CO channel. To understand the kinetic isotope effects on the reaction dynamics, we have extended our study for the C(3P) + D2CO reaction. It is seen that isotopic substitution of both the H atoms has a small reduction in the extent of ISC dynamics for the carbene formation. Our results, certainly, reveal the importance of the ketocarbene intermediate and the H + HCCO products channel as one of the major product formation channels in the title reaction, which was not reported earlier.

Metal-Induced Fast Vibrational Energy Relaxation: Quantum Nuclear Effects Captured in Diabatic Independent Electron Surface Hopping (IESH-D) Method.

In this paper, we propose a simple approach to include quantum nuclear effects in the weak electronic coupling regime within the independent electron surface hopping (IESH) method for simulating nonadiabatic dynamics near metal surfaces. Our method uses electronic states in a diabatic basis, and electronic transitions between metal and molecular states are included using Landau-Zener theory. We benchmark our novel approach using a two-state model system for which exact results (obtained from Fermi's golden rule) are available. We further investigate the effect of metallic electrons on the rate and the pathway of vibrational energy relaxation.

Ultrafast Electronic Relaxation in Aqueous [Fe(bpy)3]2+: A Surface Hopping Study.

Trajectory surface hopping simulations are performed to better understand the electronic relaxation dynamics of [Fe(bpy)3]2+ in aqueous solution. Specifically, the ultrafast relaxation from the photoexcited singlet metal-to-ligand charge-transfer (MLCT) to the metastable quintet metal-centered (MC) states is simulated through the surface hopping method, where the MLCT and MC states of [Fe(bpy)3]2+ in aqueous solution are computed by using a model electronic Hamiltonian developed previously. As a result, most of the trajectories are interpreted to show the sequential relaxation pathways via the triplet MC states, though some are the direct pathway from MLCT to the quintet MC states. Even though the triplet MC states are involved in the relaxation, the population transfer to the singlet MC ground state is very small, and the population of the quintet MC states reaches more than ∼96%, reasonably consistent with the unity quantum efficiency discussed experimentally.

An efficient decoherence scheme for fewest switches surface hopping method.

The fewest switches surface hopping method, in its original form, is known to be over-coherent. An accurate and efficient decoherence scheme is still a question of concern in the community. We propose a modification of the augmented fewest switches surface hopping (A-FSSH) scheme to make it efficient without compromising on its accuracy. We compare our proposed decoherence scheme with a recently proposed parameter-free decoherence scheme by Wang and co-workers. We show that for all model systems investigated here, our scheme is as accurate as the original A-FSSH method while at least 30 times more efficient. The decoherence scheme of Wang is more than ten times more efficient than our scheme although it does not give accurate results in the limit of large diabatic coupling.

Excited State Nonadiabatic Molecular Dynamics of Hot Electron Addition to Water Clusters in the Ultrafast Femtosecond Regime

Dissociative electron attachment (DEA) reactions of water result in the production of hydrogen atoms and hydroxide anions. This has been studied for a long time and is relatively slow in liquid water for thermalized hydrated electrons but much faster with a higher-energy electron. Here, we probe the nonadiabatic molecular dynamics after the addition of a hot electron (6-7 eV) to a neutral water cluster (H2O)n, where n = 2-12, considering the 0-100 fs time scale using the fewest switches surface hopping method, in conjunction with ab initio molecular dynamics and the Tamm-Dancoff approximation density functional theory method. The nonadiabatic DEA occurs within 10-60 fs, and with high probability, giving H + OH- above an energy threshold. This is faster than time scales estimated previously for autoionization or adiabatic DEA. The change in threshold energy with cluster size is modest, ranging from 6.6 to 6.9 eV. Dissociation on a femtosecond time scale is consistent with pulsed radiolysis experiments.

Predicting Kinetics and Dynamics of Spin-Dependent Processes

ConspectusPredicting mechanisms and rates of nonadiabatic spin-dependent processes including photoinduced intersystem crossings, thermally activated spin-forbidden reactions, and spin crossovers in metal centers is a very active field of research. These processes play critical roles in transition-metal-based and metalloenzymatic catalysis, molecular magnets, light-harvesting materials, organic light-emitting diodes, photosensitizers for photodynamic therapy, and many other applications. Therefore, accurate modeling of spin-dependent processes in complex systems and on different time scales is important for many problems in chemistry, biochemistry, and materials sciences.Nonadiabatic statistical theory (NAST) and nonadiabatic molecular dynamics (NAMD) are two complementary approaches to modeling the kinetics and dynamics of spin-dependent processes. NAST predicts the probabilities and rate constants of nonradiative transitions between electronic states with different spin multiplicities using molecular properties at only few critical points on the potential energy surfaces (PESs), including the reactant minimum and the minimum energy crossing point (MECP) between two spin states. This makes it possible to obtain molecular properties for NAST calculations using accurate but often computationally expensive electronic structure methods, which is critical for predicting the rate constants of spin-dependent processes. Alternatively, NAST can be used to study spin-dependent processes in very large complex molecular systems using less computationally expensive electronic structure methods. The nuclear quantum effects, such as zero-point vibrational energy, tunneling, and interference between reaction paths can be easily incorporated. However, the statistical and local nature of NAST makes it more suitable for large systems and slow kinetics. In contrast, NAMD explores entire PESs of interacting electronic states, making it ideal for modeling fast barrierless spin-dependent processes. Because the knowledge of large portions of PESs is often needed, the simulations require a very large number of electronic structure calculations, which limits the NAMD applicability to relatively small molecular systems and ultrafast kinetics.In this Account, we discuss our contribution to the development of the NAST and NAMD approaches for predicting the rates and mechanism of spin-dependent processes. First, we briefly describe our NAST and NAMD implementations. The NAST implementation is an extension of the transition state theory to the processes involving two crossing potential energy surfaces of different spin multiplicities. The NAMD approach includes the trajectory surface hopping (TSH) and ab initio multiple spawning (AIMS) methods. Second, we discuss several applications of NAST and NAMD to model spin-dependent processes in different systems. The NAST applicability to large complex systems is demonstrated by the studies of the spin-forbidden isomerization of the active sites of metal-sulfur proteins. Our implementation of the MECP search algorithm within the fully ab initio fragment molecular orbital method allows applying NAST to systems with thousands of atoms, such as the solvated protein rubredoxin. Applications of NAMD to ultrafast spin-dependent processes are represented by the generalized AIMS simulations utilizing the fast GPU-based TeraChem electronic structure program to gain insight into the complex photoexcited state relaxation in 2-cyclopentenone.

Nonadiabatic Coupling in Trajectory Surface Hopping: How Approximations Impact Excited-State Reaction Dynamics

Photochemical reactions are widely modeled using the popular trajectory surface hopping (TSH) method, an affordable mixed quantum-classical approximation to the full quantum dynamics of the system. TSH is able to account for nonadiabatic effects using an ensemble of trajectories, which are propagated on a single potential energy surface at a time and which can hop from one electronic state to another. The occurrences and locations of these hops are typically determined using the nonadiabatic coupling between electronic states, which can be assessed in a number of ways. In this work, we benchmark the impact of some approximations to the coupling term on the TSH dynamics for several typical isomerization and ring-opening reactions. We have identified that two of the schemes tested, the popular local diabatization scheme and a scheme based on biorthonormal wave function overlap implemented in the OpenMOLCAS code as part of this work, reproduce at a much reduced cost the dynamics obtained using the explicitly calculated nonadiabatic coupling vectors. The other two schemes tested can give different results, and in some cases, even entirely incorrect dynamics. Of these two, the scheme based on configuration interaction vectors gives unpredictable failures, while the other scheme based on the Baeck-An approximation systematically overestimates hopping to the ground state as compared to the reference approaches.

Nonadiabatic Forward Flux Sampling for Excited-State Rare Events.

We present a rare event sampling scheme applicable to coupled electronic excited states. In particular, we extend the forward flux sampling (FFS) method for rare event sampling to a nonadiabatic version (NAFFS) that uses the trajectory surface hopping (TSH) method for nonadiabatic dynamics. NAFFS is applied to two dynamically relevant excited-state models that feature an avoided crossing and a conical intersection with tunable parameters. We investigate how nonadiabatic couplings, temperature, and reaction barriers affect transition rate constants in regimes that cannot be otherwise obtained with plain, traditional TSH. The comparison with reference brute-force TSH simulations for limiting cases of rareness shows that NAFFS can be several orders of magnitude cheaper than conventional TSH and thus represents a conceptually novel tool to extend excited-state dynamics to time scales that are able to capture rare nonadiabatic events.

Semiclassical Dynamics on Machine-Learned Coupled Multireference Potential Energy Surfaces: Application to the Photodissociation of the Simplest Criegee Intermediate.

Determination of high-dimensional potential energy surfaces (PESs) and nonadiabatic couplings have always been quite challenging. To this end, machine learning (ML) models, trained with a finite set of ab initio data, allow accurate prediction of such properties. To express the PESs in terms of atomic contributions is the cornerstone of any ML based technique because it can be easily scaled to large systems. In this work, we have constructed high fidelity PESs and nonadiabatic coupling terms at the CASSCF level of ab initio data using a machine learning technique, namely, kernel-ridge regression. Additional MRCI-level calculations were carried out to assess the quality of the PESs. We use these machine-learned PESs and nonadiabatic couplings to simulate excited-state molecular dynamics based on Tully's fewest-switches surface hopping method (FSSH). FSSH is a semiclassical method in which nuclei move on the PESs due to the electrons according to the laws of classical mechanics. Nonadiabatic effects are taken into account in terms of transitions between PESs. We apply this scheme to study the O-O photodissociation of the simplest Criegee intermediate (CH2OO). The FSSH trajectories were initiated on the lowest optically bright singlet excited state (S2) and propagated along the three most important internal coordinates, namely, O-O and C-O bond distances and the COO bond angle. Some of the trajectories end up on energetically lower PESs as a result of radiationless transfer through conical intersections. All of the trajectories lead to the dissociation of the O-O bond due to the dissociative nature of the excited PESs through one of the two dissociative channels. The simulation reveals that there is about 88.4% probability of dissociation through the lower channel leading to the H2CO (X1A1) and O (1D) products, whereas there is only 11.6% probability of dissociation through the upper channel leading to H2CO (a3A″) and O (3P) products.

Efficient implementation and performance analysis of the independent electron surface hopping method for dynamics at metal surfaces.

Independent electron surface hopping (IESH) is a computational algorithm for simulating the mixed quantum-classical molecular dynamics of adsorbate atoms and molecules interacting with metal surfaces. It is capable of modeling the nonadiabatic effects of electron-hole pair excitations on molecular dynamics. Here, we present a transparent, reliable, and efficient implementation of IESH, demonstrating its ability to predict scattering and desorption probabilities across a variety of systems, ranging from model Hamiltonians to full dimensional atomistic systems. We further show how the algorithm can be modified to account for the application of an external bias potential, comparing its accuracy to results obtained using the hierarchical quantum master equation. Our results show that IESH is a practical method for modeling coupled electron-nuclear dynamics at metal surfaces, especially for highly energetic scattering events.

Decoherence and Its Role in Electronically Nonadiabatic Dynamics.

Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.

Surface hopping simulations on charge separation in an organic donor-acceptor system.

Charge separation in organic solar cells is a long-lasting and heavily debated issue. Here, we use the surface hopping method based on the Pariser-Parr-Pople (PPP) Hamiltonian and configuration interaction singles (CIS) approximation to simulate the charge separation process in an organic donor-acceptor system. The system is composed of one donor polymer chain and four acceptor polymer chains, and they are all stacked face-to-face. We let the system to relax from a photoexcited state, and then we observed that the electron is transferred from the donor chain to different acceptor chains and the hole is left on the donor chain, forming polaron pairs with different electron-hole distances. By performing statistical analysis on a number of trajectories, we found that the electron and the hole are fully separated before the system relaxes to its lowest excited state. The yield of free charges shows a significant dependence on the donor-acceptor band offset which provides the driving force for charge separation, while showing negligible dependence on the photoexcitation energy. The external electric field has a remarkable effect on the yield of free charges.

Pedagogical Overview of the Fewest Switches Surface Hopping Method.

The fewest switches surface hopping method continues to grow in popularity to capture electronic nonadiabaticity and quantum nuclear effects due to its simplicity and accuracy. Knowing the basics of the method is essential for the correct implementation and interpretation of results. This review covers the fundamentals of the fewest switches surface hopping method with a detailed discussion of the nuances such as decoherence schemes and frustrated hops and the correct approach to calculating populations. The consequences of incorrect implementation are further discussed toward calculating kinetic and thermodynamic properties. Some tips for practitioners and a step-by-step algorithm for developers are provided. Finally, some of the finer technicalities of the fewest switches surface hopping method that are buried deep in the literature are pointed out to help graduate students better appreciate this method.

Trajectory surface-hopping study of 1-pyrazoline photodissociation dynamics.

The photodissociation dynamics of 1-pyrazoline has been studied from its first excited electronic state (S1) using the Direct Dynamics Trajectory Surface-Hopping method in conjunction with Tully's fewest switches algorithm at the CASSCF(8,8)/6-31G* level of theory. After excitation of the molecule into the Franck-Condon region of the first excited state, S1, the molecule hops to the ground (S0) state quickly. The dissociation of one of the C-N bonds initially starts in the first excited state. Then, the molecule comes to the ground state (S0) via S1/S0 conical intersections, followed by complete dissociation in the ground state. Two different conical intersections are identified between the first excited singlet (S1) and the ground (S0) electronic states. One primary and three secondary dissociation channels are observed from our dynamics calculations of photodissociation of 1-pyrazoline that are in accord with the experimentally observed channels. After internal conversion to the ground electronic state (S0), the molecule dissociates to N2 and trimethylene biradical as the primary dissociation products. The trimethylene biradical then rearranges, leading to three secondary dissociation channels, N2 + cyclopropane, N2 + CH2 + C2H4, and N2 + CH3CHCH2. The major products formed from the trimethylene biradical in the secondary process is cyclopropane contributing about 78% of the overall products formation along with ∼12% propene and the rest ∼10% methylene (CH2) with ethene (C2H4).

Nonadiabatic Reactive Quenching of OH(A2Σ+) by H2: Origin of High Vibrational Excitation in the H2O Product.

The nonadiabatic dynamics of the reactive quenching channel of the OH(A2Σ+) + H2/D2 collisions is investigated with a semiclassical surface hopping method, using a recently developed four-state diabatic potential energy matrix (DPEM). In agreement with experimental observations, the H2O/HOD products are found to have significant vibrational excitation. Using a Gaussian binning method, the H2O vibrational state distribution is determined. The preferential energy disposal into the product vibrational modes is rationalized by an extended Sudden Vector Projection model, in which the h and g vectors associated with the conical intersection are found to have large projections with the product normal modes. However, our calculations did not find significant insertion trajectories, suggesting the need for further improvement of the DPEM.

Detailed Balance and Independent Electron Surface-Hopping Method: The Importance of Decoherence and Correct Calculation of Diabatic Populations.

We benchmark and improve the independent electron surface-hopping (IESH) method developed by J. C. Tully's group for nonadiabatic simulations near metal surfaces. We have incorporated decoherence within the IESH method as well as implemented a scheme for the accurate calculation of diabatic populations. We benchmark the original IESH method with the above inclusions for a model system to calculate rate constants and long-time populations. The original IESH method fails to capture the detailed balance for some of the parameters, which is corrected with the inclusion of decoherence and accurate calculation of diabatic populations. Total rate constants are well captured both within the original IESH method as well as within our modified IESH.

Transition Path Flight Times and Nonadiabatic Electronic Transitions

Transition path flight times are studied for scattering on two electronic surfaces with a single crossing. These flight times reveal nontrivial quantum effects such as resonance lifetimes and nonclassical passage times and reveal that nonadiabatic effects often increase flight times. The flight times are computed using numerically exact time propagation and compared with results obtained from the Fewest Switches Surface Hopping (FSSH) method. Comparison of the two methods shows that the FSSH method is reliable for transition path times only when the scattering is classically allowed on the relevant adiabatic surfaces. However, where quantum effects such as tunneling and resonances dominate, the FSSH method is not adequate to accurately predict the correct times and transition probabilities. These results highlight limitations in methods which do not account for quantum interference effects, and suggest that measuring flight times is important for obtaining insights from the time-domain into quantum effects in nonadiabatic scattering.

(Invited) Simulating Dynamic Excitons Via Quantum Molecular Dynamics: A Case Study in Lead Halide Perovskites

The ultrafast electronic and structural dynamics invoked by photoexcitation, i.e., “dynamic exciton” phenomena, not only are important in the context of practical applications such as solar cells, but also raise many questions from the viewpoint of fundamental science. Experimental measurement, computational simulations, and theoretical interpretation will be the three pillars for deciphering the dynamic exciton phenomena.From the viewpoint of computational simulations, molecular dynamics (MD) techniques combined with quantum chemical calculations, i.e., quantum molecular dynamics (QMD), has been the popular tool to simulate the dynamic exciton phenomena. The quantum chemical calculations, which are typically conducted on the basis of the density-functional theory (DFT), require the large computational resources and time. These have been the limiting factors for the accessible spatial and time scales by the QMD simulations.To extend the coverage of simulations to more complex, large-scale systems, we have developed efficient excited-state QMD methods that can include nonadiabatic effects. Our method combines the density-functional tight binding (DFTB) method, which is an approximate DFT, and the surface hopping method, which is a theoretical framework to incorporate the nonadiabatic effects into the QMD simulations. The method was further improved to be suitable for condensed-phase simulations explicitly including the environment, i.e., solvent, by using a “divide-and-conquer” style quantum chemical calculation technique. These theoretical framework enables us to simulate the coupled electronic–structural dynamics in excited states of systems consisting of 102–103 atoms[1,2,3].In addition, using the developed method, we conducted the real-time simulations of the ultrafast processes invoked by photoexcitation of lead iodide perovskites, which are known as the key materials for perovskite solar cells. The dissociation of the exciton into the positive and negative charge carriers was observed. Moreover, the hot carrier cooling, where the charge carriers dissipate excess energy via the electron–phonon coupling and relax to the band edges, was also tracked. Finally, the direct evidence of the polaron formation, where the structural deformation is induced by the presence of charge carriers, was observed. These results highlight the importance of the coupling between electronic and structural degrees of freedom.In the talk, recent improvements in the methodology and future perspectives will also be presented[5].