Semiclassical Surface Hopping Research Articles

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.

Tracking the nuclear movement of the carbonyl sulfide cation after strong-field ionization by time-resolved Coulomb-explosion imaging

We studied the ultrafast nuclear dynamics during the dissociation of ${\mathrm{OCS}}^{+}$ molecules using a strong IR-laser pump and probe technique in combination with the coincidence measurement. The nuclear movement is tracked by analyzing the time-dependent kinetic energy release (KER) spectra. The involved dissociation states and pathways are assigned with the help of the semiclassical Landau-Zener surface hopping calculations. The real-time bond-breaking dynamics of the ${3}^{2}\phantom{\rule{0.16em}{0ex}}A{}^{\ensuremath{'}}$ coupling to other states are observed for the two-body dissociation channel and the three-body dissociation channel but with high KER. The three-body dissociation channel with low KER is assigned to the direct breaking process from the ${3}^{2}\phantom{\rule{0.16em}{0ex}}A{}^{\ensuremath{''}}$ state. The overall agreements between the experimental and theoretical results demonstrate that the time-resolved Coulomb-explosion imaging is a valuable way to monitor the bond breaking and structural evolution of complex molecules.

Nonlinear geometric optics based multiscale stochastic Galerkin methods for highly oscillatory transport equations with random inputs

We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method, developed in Crouseilles et al. [Math. Models Methods Appl. Sci. 23 (2017) 2031–2070] for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We modify this method by introducing a new "time" variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution.

Implementation of Coherent Switching with Decay of Mixing into the SHARC Program.

Simulation of electronically nonadiabatic dynamics is an important tool for understanding the mechanisms of photochemical and photophysical processes. Two contrasting methods in which the electrons are treated quantum mechanically while the nuclei are treated classically are semiclassical Ehrenfest dynamics and trajectory surface hopping; neither method in its original form includes decoherence. Decoherence in the context of electronically nonadiabatic dynamics refers to the gradual collapse of a coherent quantum mechanical electronic state under the scrutiny of nuclear motion into a mixture of stable pointer states. This is modeled in the coherent switches with decay of mixing (CSDM) method by the decay of the off-diagonal elements of the electronic density matrix. Here, we present an implementation of CSDM in the SHARC program; a key element of the new implementation is the use of a different propagator than that used previously in the ANT program.

Simulating the Nonadiabatic Relaxation Dynamics of 4-(N,N-Dimethylamino)benzonitrile (DMABN) in Polar Solution.

The compound 4-(N,N-dimethylamino)benzonitrile (DMABN) represents the archetypal system for dual fluorescence, a rare photophysical phenomenon in which a given fluorophore shows two distinct emission bands. Despite extensive studies, the underlying mechanism remains the subject of debate. In the present contribution, we address this issue by simulating the excited-state relaxation process of DMABN as it occurs in polar solution. The potential energy surfaces for the system are constructed with the use of the additive quantum mechanics/molecular mechanics (QM/MM) method, and the coupled dynamics of the electronic wave function and the nuclei is propagated with the semiclassical fewest switches surface hopping method. The DMABN molecule, which comprises the QM subsystem, is treated with the use of the second-order algebraic diagrammatic construction (ADC(2)) method with the imposition of spin-opposite scaling (SOS). It is verified that this level of theory achieves a realistic description of the excited-state potential energy surfaces of DMABN. The simulation results qualitatively reproduce the main features of the experimentally observed fluorescence spectrum, thus allowing the unambiguous assignment of the two fluorescence bands: the normal band is due to the near-planar locally excited (LE) structure of DMABN, while the so-called "anomalous" second band arises from the twisted intramolecular charge transfer (TICT) structure. The transformation of the LE structure into the TICT structure takes place directly via intramolecular rotation, and is not mediated by another excited-state structure. In particular, the oft-discussed rehybridized intramolecular charge transfer (RICT) structure, which is characterized by a bent nitrile group, does not play a role in the relaxation process.

Photophysics of BODIPY-Based Photosensitizer for Photodynamic Therapy: Surface Hopping and Classical Molecular Dynamics.

Halogenated BODIPY derivatives are emerging as important candidates for photodynamic therapy of cancer cells due to their high triplet quantum yield. We probed fundamental photophysical properties and interactions with biological environments of such photosensitizers. To this end, we employed static TD-DFT quantum chemical calculations as well as TD-DFT surface hopping molecular dynamics on potential energy surfaces resulting from the eigenstates of the total electronic Hamiltonian including the spin-orbit (SO) coupling. Matrix elements of an effective one-electron spin-orbit Hamiltonian between singlet and triplet configuration interaction singles (CIS) auxiliary wave functions are calculated using a new code capable of dealing with singlets and both restricted and unrestricted triplets built up from up to three different and independent sets of (singlet, alpha, and beta) molecular orbitals. The interaction with a biological environment was addressed by using classical molecular dynamics (MD) in a scheme that implicitly accounts for electronically excited states. For the surface hopping trajectories, an accelerated MD approach was used, in which the SO couplings are scaled up, to make the calculations computationally feasible, and the lifetimes are extrapolated back to unscaled SO couplings. The lifetime of the first excited singlet state estimated by semiclassical surface hopping simulations is 139 ± 75 ps. Classical MD demonstrates that halogenated BODIPY in the ground state, in contrast to the unsubstituted one, is stable in the headgroup region of minimalistic cell membrane models, and while in the triplet state, the molecule relocates to the membrane interior ready for further steps of photodynamic therapy.

Surface hopping study of the photodissociation dynamics of ICN− and BrCN−

In this work the efficacy of semi-classical surface hopping approaches is investigated through studies of the photodissociation dynamics of BrCN−and ICN−. BrCN−provides a challenging situation for semi-classical approaches as excitation to the first bright state yields both Br− + CN and Br∗ + CN−products. Further, this branching is highly sensitive to the amount of rotational energy in the CN0/− fragment. The results of semi-classical and quantum mechanical descriptions of the dynamics are compared when the classical dynamics are propagated in an adiabatic and diabatic representation. The implications of the differences between the classical and quantum treatments of J=0 are also explored.

Stimulated Raman signals at conical intersections: Ab initio surface hopping simulation protocol with direct propagation of the nuclear wave function.

Femtosecond Stimulated Raman Spectroscopy (FSRS) signals that monitor the excited state conical intersections dynamics of acrolein are simulated. An effective time dependent Hamiltonian for two C-H vibrational marker bands is constructed on the fly using a local mode expansion combined with a semi-classical surface hopping simulation protocol. The signals are obtained by a direct forward and backward propagation of the vibrational wave function on a numerical grid. Earlier work is extended to fully incorporate the anharmonicities and intermode couplings.

A variational surface hopping algorithm for the sub-Ohmic spin-boson model

The Davydov D1 ansatz, which assigns individual bosonic trajectories to each spin state, is an efficient, yet extremely accurate trial state for time-dependent variation of the sub-Ohmic spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. A surface hopping algorithm is developed employing the Davydov D1 ansatz to study the spin dynamics with a sub-Ohmic bosonic bath. The algorithm takes into account both coherent and incoherent dynamics of the population evolution in a unified manner, and compared with semiclassical surface hopping algorithms, hopping rates calculated in this work follow more closely the Marcus formula.

Surface hopping dynamics of direct trans → cis photoswitching of an azobenzene derivative in constrained adsorbate geometries

With ongoing miniaturization of electronic devices, the need for individually addressable, switchable molecules arises. An example are azobenzenes on surfaces which have been shown to be switchable between trans and cis forms. Here, we examine the "direct" (rather than substrate-mediated) channel of the trans → cis photoisomerization after ππ∗ excitation of tetra-tert-butyl-azobenzene physisorbed on surfaces mimicking Au(111) and Bi(111), respectively. In spirit of the direct channel, the electronic structure of the surface is neglected, the latter merely acting as a rigid platform which weakly interacts with the molecule via Van-der-Waals forces. Starting from thermal ensembles which represent the trans-form, sudden excitations promote the molecules to ππ∗-excited states which are non-adiabatically coupled among themselves and to a nπ∗-excited and the ground state, respectively. After excitation, relaxation to the ground state by internal conversion takes place, possibly accompanied by isomerization. The process is described here by "on the fly" semiclassical surface hopping dynamics in conjunction with a semiempirical Hamiltonian (AM1) and configuration-interaction type methods. It is found that steric constraints imposed by the substrate lead to reduced but non-vanishing, trans → cis reaction yields and longer internal conversion times than for the isolated molecule. Implications for recent experiments for azobenzenes on surfaces are discussed.

Electronic coherence within the semiclassical field-induced surface hopping method: strong field quantum control in K2

We demonstrate that the semiclassical field-induced surface hopping (FISH) method (Mitrićet al., Phys. Rev. A: At., Mol., Opt. Phys., 2009, 79, 053416.) accurately describes the selective coherent control of electronic state populations. With the example of the strong field control in the potassium dimer using phase-coherent double pulse sequences, we present a detailed comparison between FISH simulations and exact quantum dynamics. We show that for short pulses the variation of the time delay between the subpulses allows for a selective population of the desired final state with high efficiency. Furthermore, also for pulses of longer time duration, when substantial nuclear motion takes place during the action of the pulse, optimized pulse shapes can be obtained which lead to selective population transfer. For both types of pulses, the FISH method almost perfectly reproduces the exact quantum mechanical electronic population dynamics, fully taking account of the electronic coherence, and describes the leading features of the nuclear dynamics accurately. Due to the significantly higher computational efficiency of FISH as a trajectory-based method compared to full quantum dynamics simulations, this offers the possibility to theoretically investigate control experiments on realistic systems including all nuclear degrees of freedom.

New Implementation of Semi-classical Dynamic Simulation on the Photoisomerization ofcis- and trans-Isomers of Free Stilbene

New implementation of semi-classical trajectory surface hopping dynamic simulation has been developed and applied to the photoisomerization of cis- and trans- isomers on the gas phase. This method not only uses the exponential model to the modification of the originally analytical non-adiabatic transition probability formula, but also involves the con- strained Hamiltonian system into the constrained molecular dynamic simulation. Two-dimensional potential energy surfaces of ground S0 and excited S1 states are constructed analytically fitting to ab initio calculations in terms of torsion angle and one dihedral angle around the central ethylenic C=C bond as variables, and the other internal coordinates are all fixed at configuration of one-bond flip conical intersection. The analytical PESs are quite accurate and the mean absolute error is less than 2.4 kcalmol -1 , and much less than 1.0 kcalmol -1 around conical intersection region. A straight seam line is found on potential energy surfaces that simply separates the cis-area with the trans-area. The constrained Hamiltonian system is em- ployed to run trajectories in the Cartesian coordinate system and surface hopping in terms of the two internal dihedral angles. Typical trajectories are found in which the torsion angle changes monotonically for both cis- to trans- and trans- to cis- isomerizations. This is an exact picture of one-bond flip mechanism of photoisomerization around the conical intersection. Quantum yield for trans- to cis- isomerization is simulated as 60.45% in very good agreement with experimental value 55.0%, while quantum yield for cis- to trans- isomerization is simulated as 42.3% in comparison with experimental value 35.0%. As the S1 energy in local minimum of cis-area is higher than that in trans-area, and thus cis- to trans- isomerization is quite possible to access to another Hula-Twist conical intersection. These simulation results demonstrate that the computed cumulative quantum yield and reaction mechanism are consistent with the previously experimental and theoretical results. This means that the present trajectory surface hopping method would be good at the dynamic simulation on the large system with or without constraint Hamiltonian in comparison with the quantum molecular dynamics. Keywords new implementation of semi-classical dynamic simulation; constrained Hamiltonian system; Zhu-Nakamura theory; isomerization of cis- and trans-stilbene; two dimensional analytical potential energy surfaces

A justification for the use of approximate transition amplitudes in semiclassical surface hopping

Recent work has shown that a certain surface hopping form of the wave function is capable of obtaining highly accurate transition probabilities for nonadiabatic problems. It has also been found that it is necessary to include hops in classically forbidden regions in order to obtain this level of accuracy at low energies. The amplitude for the hops in this surface hopping expansion of the wave function has the typical p−1/2 semiclassical divergence at the turning points in the classical motion. While this singularity is an integrable divergence, the divergent behavior complicates the numerical evaluation of the integrals over hopping points that is present in the surface hopping expressions. Numerical evidence has shown that only small errors are incurred at most energies if these singular hopping amplitudes are replaced with a nonsingular approximation. This agreement is surprising, since the exact and approximate amplitudes differ greatly in the turning point region, and this region is expected to make important contributions to the transition probability at low energies. A numerical analysis is presented in this work that provides a justification as to why this numerically useful approximation works as well as it does.

Surface Hopping Excited-State Dynamics Study of the Photoisomerization of a Light-Driven Fluorene Molecular Rotary Motor.

We report a theoretical study of the photoisomerization step in the operating cycle of a prototypical fluorene-based molecular rotary motor (1). The potential energy surfaces of the ground electronic state (S0) and the first singlet excited state (S1) are explored by semiempirical quantum-chemical calculations using the orthogonalization-corrected OM2 Hamiltonian in combination with a multireference configuration interaction (MRCI) treatment. The OM2/MRCI results for the S0 and S1 minima of the relevant 1-P and 1-M isomers and for the corresponding S0 transition state are in good agreement with higher-level calculations, both with regard to geometries and energetics. The S1 surface is characterized at the OM2/MRCI level by locating two S0-S1 minimum-energy conical intersections and nearby points on the intersection seam and by computing energy profiles for pathways toward these MECIs. Semiclassical Tully-type trajectory surface hopping (TSH) simulations with on-the-fly OM2/MRCI calculations are carried out to study the excited-state dynamics after photoexcitation to the S1 state. Fast relaxation to the ground state is observed through the conical intersection regions, predominantly through the lowest-energy one with a strongly twisted central C═C double bond and pyramidalized central carbon atom. The excited-state lifetimes for the direct and inverse photoisomerization reactions (1.40 and 1.79 ps) and the photostationary state ratio (2.7:1) from the TSH-OM2 simulations are in good agreement with the available experimental data (ca. 1.7 ps and 3:1). Excited-state lifetimes, photostationary state ratio, and dynamical details of the TSH-OM2 simulations also agree with classical molecular dynamics simulations using a reparametrized optimized potentials for liquid simulations (OPLS) all-atom force field with ad-hoc surface hops at predefined conical intersection points. The latter approach allows for a more extensive statistical sampling.

Simulation of laser-induced coupled electron-nuclear dynamics and time-resolved harmonic spectra in complex systems

We present a theoretical approach for the simulation of time-resolved harmonic spectra, including the effect of nuclear dynamics, which is applicable to complex systems involving many nuclear degrees of freedom. The method is based on the combination of our semiclassical field-induced surface hopping approach for the treatment of laser-induced nuclear dynamics with the time-dependent density functional theory for electron dynamics. We apply our method to the simulation of ultrafast nonadiabatic dynamics and time-resolved harmonic spectra in small silver clusters (Ag${}_{2}$ and Ag${}_{8}$), which exhibit discrete molecularlike electronic transitions. We demonstrate that the harmonic signal is highly sensitive to the nuclear dynamics and thus can be used as a probe of coupled electron-nuclear dynamics, which is complementary to common pump-probe methods such as time-resolved photoelectron spectroscopy. Our simulations allowed us also to determine the mechanism and the time scale of nonradiative relaxation in the ``magic'' Ag${}_{8}$ cluster and have provided a fundamental insight into ultrafast dynamics of metal nanoclusters in the size regime where ``each atom counts.'' The excited-state dynamics of Ag${}_{8}$ involves an isomerization process from the initial structure with ${T}_{\mathrm{d}}$ symmetry to the quadratic antiprism structure with ${D}_{4\mathrm{d}}$ symmetry which takes place on a time scale of $~$600 fs and is clearly identified in a time-resolved harmonic signal. Our theoretical approach is generally applicable for the prediction of time-resolved harmonic spectra in complex systems with many nuclear degrees freedom and should serve to stimulate new ultrafast experiments utilizing harmonic signals as a probe for nonadiabatic processes in molecular systems.

Semiclassical dynamics simulations of charge transport in stacked π-systems

Charge transfer processes within stacked π-systems were examined for the stacked ethylene dimer radical cation with inclusion of a bridge containing up to three formaldehyde molecules. The electronic structure was treated at the complete active space self-consistent field and multireference configuration interaction levels. Nonadiabatic interactions between electronic and nuclear degrees of freedom were included through semiclassical surface hopping dynamics. The processes were analyzed according to fragment charge differences. Static calculations explored the dependence of the electronic coupling and on-site energies on varying geometric parameters and on the inclusion of a bridge. The dynamics simulations gave the possibility for directly observing complex charge transfer and diabatic trapping events.

Using semiclassical surface hopping for coupled partial wave calculations on systems with non-spherically symmetric potentials

It is shown that a semiclassical surface hopping (SH) approach provides a simple and efficient method for scattering calculations with non-spherically symmetric potentials. The calculations are performed by expanding the wave function in an angular momentum state basis. Since the potential is not spherically symmetric, the different angular states are coupled. The semiclassical SH method, which is typically used for problems with coupled electronic states, can, in principle, be employed for any coupled state problem. The particular SH method employed is known to provide highly accurate results for coupled electronic state problems. The method is tested on model two angular state problems using potential surfaces and couplings arising from a non-spherically symmetric scattering problem. The results for these model problems are in excellent agreement with exact quantum calculations. Full calculations, which are converged with regard to the number of angular basis states, are also performed for the non-spherically symmetric problem. It is shown that an approximation to the surface hopping amplitudes that simplifies the numerical implementation of the method provides results in excellent agreement with the full surface hopping calculation.

Ultrafast nonadiabatic photodissociation dynamics of F2 in solid Ar

We explore the ultrafast spin-flip dynamics in a diatomic molecule imbedded in a rare gas matrix using the combination of a quantum mechanical and a semiclassical surface hopping method. Specifically, we investigate (1) the extent to which the phenomenon of electronically-localized eigenstates in strongly-coupled manifolds survives in the presence of rapid decay and a multitude of electronically coupled states; (2) the ability of the surface hopping method to predict the short time dynamics; and (3) the time range over which frozen lattice models are valid. Our results point to the active role played by a large number of coupled electronic states in the F2/Ar dynamics while substantiating our confidence in the validity of the popular surface hopping approach for the system considered.

A semiclassical model for the calculation of nonadiabatic transition probabilities for classically forbidden transitions

A semiclassical surface hopping model is presented for the calculation of nonadiabatic transition probabilities for the case in which the avoided crossing point is in the classically forbidden regions. The exact potentials and coupling are replaced with simple functional forms that are fitted to the values, evaluated at the turning point in the classical motion, of the Born-Oppenheimer potentials, the nonadiabatic coupling, and their first few derivatives. For the one-dimensional model considered, reasonably accurate results for transition probabilities are obtained down to around 10(-10). The possible extension of this model to many dimensional problems is discussed. The fact that the model requires only information at the turning point, a point that the trajectories encounter would be a significant advantage in many dimensional problems over Landau-Zener type models, which require information at the avoided crossing seam, which is in the forbidden region where the trajectories do not go.

A time-dependent density-functional approach to nonadiabatic electron-nucleus dynamics: formulation and photochemical application

To study nonadiabatic dynamics of the electrons and nuclei, the quantum chemical wavefunction methods have often been invoked to compute the nonadiabatic couplings (NACs), but time-dependent density functional theory (TD-DFT) can provide a formally exact alternative approach when the ground and one excited electronic states are concerned. Based on the density response scheme to compute the NAC vectors [J. Chem. Phys., 2007, 127, 064103], herein presented are a full quantum wave packet and a semi-classical surface hopping approach to the nonadiabatic chemical reactions for the electronically ground and excited states. The adiabatic local density approximation (ALDA) was used here but, contrary to previous simulations based on DFT or TD-DFT, no further approximations were made for the electrons. With those approaches we could successfully describe the photochemical syn-anti isomerization dynamics of a formaldimine molecule (CH2=NH) and investigate the dissipation effects with use of a Langevin dynamics scheme. These simulations demonstrated an important role played by the dissipation and suggested that accurately modeling the dissipation is the next step towards a truly ab initio prediction.