Trajectory Surface Hopping Method Research Articles

Adiabatic and nonadiabatic dissociation of ethyl radical

Direct ab initio molecular dynamics using the trajectory surface hopping method with Tully's fewest switches simulates the photodissociation dynamics of ethyl radical, C(2)H(5), following electronic excitation to the A-state. Nonadiabatic dissociation dominates and produces ground state ethylene and fast hydrogen atoms with an anisotropic angular distribution. Surface hopping also generates hot ground state ethyl radicals followed ultimately by unimolecular dissociation to C(2)H(4)+H. The calculated excited state lifetime and the product recoil energy distribution obtained from an ensemble of trajectories are consistent with previous experiments and suggest that a strictly nonadiabatic mechanism can account for nonradiative decay. This process is in competition with adiabatic dissociation producing electronically excited state ethylene and H, a dissociation channel that has not yet been experimentally observed. The branching ratio between adiabatic and nonadiabatic dissociation pathways depends sensitively on the quality of the potential energy surfaces. At the multireference configuration interaction with singles and doubles level of theory, 15% of all trajectories dissociate adiabatically.

Photochemical deactivation pathways of the Ã-state allyl radical

Ab initio direct molecular dynamics with trajectory surface hopping methods simulates the photochemical deactivation pathways of the allyl radical, C(3)H(5), following electronic excitation to the A-state. The electronically nonadiabatic dynamics mediated by two conical intersections produces predominantly hot ground state allyl radicals along both the disrotatory and conrotatory photochemical deactivation pathways with a near synchronous rotation of the terminal methylene groups. The electrocyclic transformation of the allyl radical to the cyclopropyl radical is a minor channel accounting for 8% of all trajectories with 98% of them following the disrotatory pathway.

Algorithmic decoherence time for decay-of-mixing non–Born–Oppenheimer dynamics

The performance of an analytical expression for algorithmic decoherence time is investigated for non-Born-Oppenheimer molecular dynamics. There are two terms in the function that represents the dependence of the decoherence time on the system parameters; one represents decoherence due to the quantum time-energy uncertainty principle and the other represents a back reaction from the decoherent force on the classical trajectory. We particularly examine the question of whether the first term should dominate. Five one-dimensional two-state model systems that represent limits of multidimensional nonadiabatic dynamics are designed for testing mixed quantum-classical methods and for comparing semiclassical calculations with exact quantum calculations. Simulations are carried out with the semiclassical Ehrenfest method (SE), Tully's fewest switch version (TFS) of the trajectory surface hopping method, and the decay-of-mixing method with natural switching, coherent switching (CSDM), and coherent switching with reinitiation (CSDM-D). The CSDM method is demonstrated to be the most accurate method, and it has several desirable features: (i) It behaves like the representation-independent SE method in the strong nonadiabatic coupling regions; (ii) it behaves physically like the TFS method in noninteractive region; and (iii) the trajectories are continuous with continuous momenta. The CSDM method is also demonstrated to balance coherence well with decoherence, and the results are nearly independent of whether one uses the adiabatic or diabatic representation. The present results provide new insight into the formulation of a physically correct decoherence time to be used with the CSDM method for non-Born-Oppenheimer molecular dynamic simulations.

The three-dimensional nonadiabatic dynamics calculation of DH2+ and HD2+ systems by using the trajectory surface hopping method based on the Zhu–Nakamura theory

A theoretical investigation on the nonadiabatic processes of the full three-dimensional D(+)+H(2) and H(+)+D(2) reaction systems has been performed by using trajectory surface hopping (TSH) method based on the Zhu-Nakamura (ZN) theory. This ZN-TSH method refers to not only classically allowed hops but also classically forbidden hops. The potential energy surface constructed by Kamisaka et al. is employed in the calculation. A new iterative method is proposed to yield the two-dimensional seam surface from the topography of the adiabatic potential surfaces, in which the inconvenience of directly solving the first-order partial differential equation is avoided. The cross sections of these two systems are calculated for three competing channels of the reactive charge transfer, the nonreactive charge transfer, and the reactive noncharge transfer, for ground rovibrational state of H(2) or D(2). Also, this study provides reaction probabilities of these three processes for the total angular momentum J=0 and ground initial vibrational state of H(2) or D(2). The calculated results from ZN-TSH method are in good agreement with the exact quantum calculations and the experimental measurements.

Trajectory Surface Hopping Study of the O(3P) + Ethylene Reaction Dynamics

Spin-orbit coupling (SOC) induced intersystem crossing (ISC) has long been believed to play a crucial role in determining the product distributions in the O(3P) + C2H4 reaction. In this paper, we present the first nonadiabatic dynamics study of the title reaction at two center-of-mass collision energies: 0.56 eV, which is barely above the H-atom abstraction barrier on the triplet surface, and 3.0 eV, which is in the hyperthermal regime. The calculations were performed using a quasiclassical trajectory surface hopping (TSH) method with the potential energy surface generated on the fly at the unrestricted B3LYP/6-31G(d,p) level of theory. To simplify our calculations, nonadiabatic transitions were only considered when the singlet surface intersects the triplet surface. At the crossing points, Landau-Zener transition probabilities were computed assuming a fixed spin-orbit coupling parameter, which was taken to be 70 cm-1 in most calculations. Comparison with a recent crossed molecular beam experiment at 0.56 eV collision energy shows qualitative agreement as to the primary product branching ratios, with the CH3 + CHO and H + CH2CHO channels accounting for over 70% of total product formation. However, our direct dynamics TSH calculations overestimate ISC so that the total triplet/singlet ratio is 25:75, compared to the observed 43:57. Smaller values of SOC reduce ISC, resulting in better agreement with the experimental product relative yields; we demonstrate that these smaller SOC values are close to being consistent with estimates based on CASSCF calculations. As the collision energy increases, ISC becomes much less important and at 3.0 eV, the triplet to singlet branching ratio is 71:29. As a result, the triplet products CH2 + CH2O, H + CH2CHO and OH + C2H3 dominate over the singlet products CH3 + CHO, H2 + CH2CO, etc.

Simulation of the photodeactivation of formamide in the nO-π∗ and π-π∗ states: An ab initio on-the-fly surface-hopping dynamics study

The short-time photodynamics (1 ps) of formamide in its low-lying singlet excited n(O)-pi(*) and pi-pi(*) states have been investigated by the direct trajectory surface-hopping method based on multiconfigurational ab initio calculations. The simulations showed that in both states, the primary deactivation process is C-N bond dissociation. In the ground state, the energy is transferred to (a) translational motion of the HCO and NH(2) fragments, (b) additional C-H dissociation from the vibrationally hot HCO fragment, or (c) formation of NH(3) and CO. In addition to the C-N dissociation pathway, C-O bond fission is found to be an additional primary deactivation path in the pi-pi(*) dynamics. From fractional occupations of trajectories, lifetimes of formamide were estimated: tau(S(1))=441 fs and tau(S(2))=66 fs.

Ab initio Study of Ultrafast Photochemical Reaction Dynamics of Phenol Blue

Phenol blue (PB) is a primary skeletal structure part of indoaniline dyes and well-known as a solvatochromic dye. It has been recently observed by pump-probe (PP) transient absorption measurements that PB shows ultrafast ground state recovery within a few hundred femtoseconds after photoexcitation. In this work, the ultrafast photochemical reaction mechanism of PB has been investigated using direct ab initio (CASSCF) nonadiabatic molecular dynamics with the trajectory surface hopping (TSH) method. The swarm of trajectories starting from the S1 Franck-Condon (FC) point has mostly shown surface hops (nonadiabatic transitions) from the S1 state to the S0 state at 110-120 fs in the vicinity of an S1/S0 conical intersection and after decay to the S0 state bifurcated into two (Reverse and Forward) directions with almost the same branching ratio and reached the vicinity of the S0 minimum energy point at 200-300 fs, which is in good agreement with the fast time component of the ground state recovery in the PP measurements. After reaching the vicinity of the S0 minimum energy point, the trajectories showed a coherent vibration of bending motion between quinoneimine and aniline rings with a low frequency of 43 cm-1, which presumably corresponds to a coherently photoexcitation-induced vibrational mode with a low frequency recently observed by the PP measurements.

Critical appraisal of the fewest switches algorithm for surface hopping

In this paper the authors address the problem of internal consistency in trajectory surface hopping methods, i.e., the requirement that the fraction of trajectories running on each electronic state equals the probabilities computed by the electronic time-dependent Schrodinger equation, after averaging over all trajectories. They derive a formula for the hopping probability in Tully's "fewest switches" spirit that would yield a rigorously consistent treatment. They show the relationship of Tully's widely used surface hopping algorithm with the "exact" prescription that cannot be applied when running each trajectory independently. They also bring out the connection of the consistency problem with the coherent propagation of the electronic wave function and the artifacts caused by coherent Rabi-type oscillations of the state probabilities in weak coupling regimes. A real molecule (azobenzene) and two ad hoc models serve as examples to illustrate the above theoretical arguments. Following a proposal by Truhlar's group [Zhu et al., J. Chem. Phys. 121, 7658 (2004) Zhu et al., J. Chem. Theory Comput. 1, 527 (2005)], they apply a decoherence correction to the state probabilities, in conjunction with Tully's algorithm, and they obtain satisfactory results in terms of internal consistency and of agreement with the outcomes of quantum wave packet calculations.

Excited state dynamics with the direct trajectory surface hopping method: azobenzene and its derivatives as a case study

We review the recent studies of the photoisomerization dynamics of azobenzene and its derivatives by surface hopping simulations based on semiempirical potential energy surfaces. We examine the ability of semiclassical methods to predict the excited state dynamics and to reproduce transient spectroscopic signals that constitute the most direct experimental evidence in this field. We show that the available simulation methods yield a deep insight into the mechanism of photochemical reactions and excited state decay, and a fairly good quantitative agreement with experimental findings. Probably the most important technical improvements we can envisage concern the surface hopping algorithm and the usage of ab initio data in the simulation of transient spectra. Concerning azobenzene, our results show that the isomerization mechanism is torsion of the N=N double bond, both by n → π* and by π → π* excitation. The influence of the solvent and the findings of some recent femtochemistry experiments deserve further work to be fully interpreted.

Dynamics of Nonadiabatic Chemical Reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.

Generalized trajectory surface hopping method based on the Zhu-Nakamura theory

We present a generalized formulation of the trajectory surface hopping method applicable to a general multidimensional system. The method is based on the Zhu-Nakamura theory of a nonadiabatic transition and therefore includes the treatment of classically forbidden hops. The method uses a generalized recipe for the conservation of angular momentum after forbidden hops and an approximation for determining a nonadiabatic transition direction which is crucial when the coupling vector is unavailable. This method also eliminates the need for a rigorous location of the seam surface, thereby ensuring its applicability to a wide class of chemical systems. In a test calculation, we implement the method for the DH(2) (+) system, and it shows a remarkable agreement with the previous results of C. Zhu, H. Kamisaka, and H. Nakamura, [J. Chem. Phys. 116, 3234 (2002)]. We then apply it to a diatomic-in-molecule model system with a conical intersection, and the results compare well with exact quantum calculations. The successful application to the conical intersection system confirms the possibility of directly extending the present method to an arbitrary potential of general topology.

Dynamics of the O(1D) D2 reaction: A comparison between crossed molecular beam experiments and quasiclassical trajectory calculations on the lowest three potential energy surfaces

This paper describes crossed beam experiments and quasiclassical trajectory (QCT) calculations for the 'insertion' reaction O(1D)+D2 at a collision energy (Ec=25.9 kJ mol-1) much higher than the calculated barrier (∼8.4 kJ mol-1) for the competitive 'abstraction' mechanism, which takes place along the excited state potential energy surfaces (PES). Adiabatic QCT calculations were carried out on the ground 11A' and first excited 11A'' PESs developed by Dobbyn and Knowles. Non-adiabatic contributions from the excited 21A' PES to the reaction were considered by means of a trajectory surface hopping method. QCT calculations were also performed at Ec=22.2 kJ mol-1 to compare with previous experimental results. Excellent agreement was found between experiment and QCT predictions at Ec=22.2 kJ mol-1, while at the higher Ec of 25.9 kJ mol-1 only a qualitative agreement was noted. In all cases, the comparison was significantly improved with respect to QCT calculations on a previous version of the ground state PES.

The triplet channel in the photodissociation of ozone in the Hartley band: Classical trajectory surface hopping analysis

The triplet channel in the photodissociation of ozone in the Hartley band, O3 + hv-->O(3P) + O2(3sigma), is investigated by means of a classical trajectory surface hopping method using ab initio diabatic potential energy surfaces for the B and the R states. Because of the strong recoil in the R state along the breaking O-O bond, O2(3sigma) is produced with a high rotational energy. The nonadiabatic transition probability depends markedly on the coordinate along the crossing seam. As a consequence a unique correlation is found between the internuclear geometry at the crossing and the final vibrational state of O2(3sigma). The calculated distribution of the translational energy is in good accord with the measured distribution.

Coherent switching with decay of mixing: an improved treatment of electronic coherence for non-Born-Oppenheimer trajectories.

The self-consistent decay-of-mixing (SCDM) semiclassical trajectory method for electronically nonadiabatic dynamics is improved by modifying the switching probability that determines the instantaneous electronic state toward which the system decoheres. This method is called coherent switching with decay of mixing (CSDM), and it differs from the previously presented SCDM method in that the electronic amplitudes controlling the switching of the decoherent state are treated fully coherently in the electronic equations of motion for each complete passage through a strong interaction region. It is tested against accurate quantum mechanical calculations for 12 atom-diatom scattering test cases. Also tested are the SCDM method and the trajectory surface hopping method of Parlant and Gislason that requires coherent passages through each strong interaction region, and which we call the "exact complete passage" trajectory surface hopping (ECP-TSH) method. The results are compared with previously presented results for the fewest switches with time uncertainty and Tully's fewest switches (TFS) surface hopping methods and the semiclassical Ehrenfest method. We find that the CSDM method is the most accurate of the semiclassical trajectory methods tested. Including coherent passages improves the accuracy of the SCDM method (i.e., the CSDM method is more accurate than the SCDM method) but not of the trajectory surface hopping method (i.e., the ECP-TSH method is not more accurate on average than the TFS method).

Importance of Intersystem Crossing in the S(3P, 1D) + H2 → SH + H Reaction

A “mixed” representation approach in conjunction with a trajectory surface-hopping method is used to study intersystem crossing effects in the S + H2 reaction. These calculations are based on high-quality potential surfaces that we have determined for the two lowest triplet states of SH2 and globally determined spin−orbit coupling matrix elements that are obtained from CASSCF calculations. A previously determined surface for the lowest singlet state (Ho, T.-S.; Hollebeek, T.; Rabitz, H.; Chao, S. D.; Skodje, R. T.; Zyubin, A. S.; Mebel, A. M. J. Chem. Phys. 2002, 116, 4124) is also used. We find that in contrast to the O(3P) + H2 reaction, which we studied previously at the same level, there is significant intersystem crossing in the S(3P) + H2 reaction. In particular, for the reaction starting from triplet S + H2 close to the threshold, the dominant mechanism involves intersystem crossing to the singlet state prior to encountering the triplet barrier, and as a result, the thermal rate constant at low tem...

Non-Born-Oppenheimer trajectories with self-consistent decay of mixing.

A semiclassical trajectory method, called the self-consistent decay of mixing (SCDM) method, is presented for the treatment of electronically nonadiabatic dynamics. The SCDM method is a modification of the semiclassical Ehrenfest (SE) method (also called the semiclassical time-dependent self-consistent-field method) that solves the problem of unphysical mixed final states by including decay-of-mixing terms in the equations for the evolution of the electronic state populations. These terms generate a force, called the decoherent force (or dephasing force), that drives the electronic component of each trajectory toward a pure state. Results for several mixed quantum-classical methods, in particular the SCDM, SE, and natural-decay-of-mixing methods and several trajectory surface hopping methods, are compared to the results of accurate quantum mechanical calculations for 12 cases involving five different fully dimensional triatomic model systems. The SCDM method is found to be the most accurate of the methods tested. The method should be useful for the simulation of photochemical reactions.

Unified Semiclassical Theory for the Nonadiabatic Transition

AbstractUnified semiclassical theory for general two‐state nonadiabatic transition is discussed by demonstrating an important quantity that represents a type of nonadiabatic transition. This quantity is essentially defined by the rotation angle which determines transformation between diabatic and adiabatic representation, and it runs from unity to infinity continuously. The unified semiclassical formula is numerically demonstrated, especially in the high energy range where the transition probability is sensitive to the type of nonadiabatic transition. The present semiclassical theory can be incorporated with the framework of the Tully's fewest‐switches trajectory surface hopping method to deal with nonadiabatic transitions in multi‐dimensional systems.

Trajectory surface hopping study of the C + CH reaction

The influence of electronically nonadiabatic transitions in the C(3Pg) + CH(X 2Π) → C2(X 1Σ+g, a 3Πu) + H(2Sg) reaction is investigated by using Tully's fewest-switches version of the trajectory surface hopping method. A diabatic model of the first two 2A′ potential energy surfaces coupled by a conical intersection is used. The diatomic CH has the internal state (ν = 0, j = 0) and batches of 20 000 trajectories are computed for four collision energies, E = 0.1, 0.3, 0.5 and 0.7 eV. We find that the reaction dynamics does not exhibit a statistical character, despite the existence of deep wells along the reaction path. Only the distribution of scattering angle shows a good agreement between trajectories and phase space theory results. A strong excess of vibrational energy is disposed on both electronic products C2(X 1Σ+g) and C2(a 3Πu), correlated with a lack of recoil energy. With all trajectories starting on a single potential surface, we obtain an electronic branching ratio X : a close to 2 : 3, only slightly dependent on the collision energy.

Quantum Scattering Calculations of the O(1D) + N2(XΣg+) → O(3P) + N2(X1Σg+) Spin-Forbidden Electronic Quenching Collision

Three-dimensional quantum scattering calculations have been carried out for the O(1D) + N2 → O(3P) + N2 spin-forbidden electronic quenching process using a simplified collision model, in which only the lowest singlet surface and one triplet surface are taken into account. The standard close-coupling technique has been used to obtain nonadiabatic transition probabilities, and the coupled-state approximation was applied to calculate the total quenching cross section. Previously developed analytical potential energy surfaces and the spin−orbit coupling element have been employed. The results of the close-coupling calculations have been compared to the results of the quasiclassical trajectory surface hopping method. Two versions of the method have been used; one uses Tully's fewest switches algorithm, and the other is the hopping method of Preston and Tully. It has been found that both of the trajectory surface hopping methods give too large quenching probabilities compared to those of the quantum results, in which the quenching dynamics is exclusively resonance-dominated. Detailed analyses of the quantum results show that a curve crossing picture cannot be employed to describe the present nonadiabatic collision. The calculated quenching cross sections have been also compared to those of experimental data as well as previous theoretical results.

Li + Li2 Dissociation Reaction Using the Self-Consistent Potential and Trajectory Surface Hopping Methods

Self-consistent potential and trajectory surface hopping methods have been applied to study the Li + Li2 dissociation reaction. Both methods fall into the classical trajectory methodology, with batches of 5000 trajectories being run over the translational energy range 25 ≤ Etr ≤ 100 kcal mol-1 keeping the internal state of Li2 fixed at (v = 0, j = 10). The effect of vibrational excitation has also been studied by running additional sets of trajectories for Etr = 25 kcal mol-1 with (v = 10, j = 10) and (v = 20, j = 10). All dissociative cross sections have been calculated using realistic double many-body expansion potential energy surfaces. The importance of nonadiabatic effects is investigated.