Classical Trajectory Calculations Research Articles

Mixed, quantum-classical description of electron density transfer in the collision process

In this work, we investigate an ion-atom model describing the time-dependent evolution of electron density during the collision. For a S 3 + − H system, numerical simulations are based on classical trajectory calculations, and the electron density behaviour is described with the time-dependent Schrödinger equation. We apply the finite difference method to obtain quantitative insights into the charge transfer dynamics, providing detailed information about the spatial and temporal evolution of the collision process. The results are given for representative examples of the collision, from eV to keV range of energies, in head-on collision as well as for different values of impact parameter. A validity and precision of the proposed model and interpretation of the particle collision in terms of eigenstates are also discussed.

HO2•$_{2}^{\bullet }$+O3 → OH•+2O2 reaction: A potential source of vibrationally hot OH radicals in the atmosphere

AbstractIn the present work, using on‐the‐fly classical trajectory calculations along with quantum chemical computation, we have shown that HO+O3→ OH•+2O2 reaction can be a potential source of the vibrationally excited OH radical. The investigation suggests that OH radical will be majorly produced in ν=1 and ν=2 states. We have also shown that the vibrationally hot OH radical is key in interpreting the observed branching fraction of 16OH• in the gas phase experiment of Nelson and Zahniser (J. Phys. Chem. 1994, 98, 2101–2104). Lastly, we have discussed the atmospheric implications of title reaction by comparing it with other chemical reactions known to produce hot OH radical in the atmosphere.

Torsional Wave-Packet Dynamics in 2-Fluorobiphenyl Investigated by State-Selective Ionization-Detected Impulsive Stimulated Raman Spectroscopy.

We report the creation and observation of vibrational wave packets pertinent to torsional motion in a biphenyl derivative in its electronic ground-state manifold. Adiabatically cooled molecular samples of 2-fluorobiphenyl were irradiated by intense nonresonant ultrashort laser pulses to drive impulsive stimulated Raman excitation of torsional motion. Spectral change due to the nonadiabatic vibrational excitation is probed in a state-selective manner using resonance-enhanced two-photon ionization through the S1 ← S0 electronic transition. The coherent nature of the excitation was exemplified by adopting irradiation with a pair of pump pulses: observed signals for excited torsional levels exhibit oscillatory variations against the mutual delay between the pump pulses due to wave-packet interference. By taking the Fourier transform of the time course of the signals, energy intervals among torsional levels with v = 0-3 were determined and utilized to calibrate a density functional theory (DFT)-calculated torsional potential-energy function. Time variation of populations in the excited torsional levels was assessed experimentally by measuring integrated intensities of the corresponding transitions while scanning the delay. Early time enhancement of the population (up to ∼2 ps) and gradual degradation of coherence (within ∼20 ps) appears. To explain the observed distinctive features, we developed a four-dimensional (4D) dynamical calculation in which one-dimensional (1D) quantum-mechanical propagation of the torsional motion was followed by solving the time-dependent Schrödinger equation, whereas three-dimensional (3D) molecular rotation was tracked by classical trajectory calculations. This hybrid approach enabled us to reproduce experimental results at a reasonable computational cost and provided a deeper insight into rotational effects on vibrational wave-packet dynamics.

Ozone Photodissociation in the Singlet Channel at 226 nm.

We report the rotational state distribution and vector correlations of the O2(a 1Δg, v = 0) fragments arising from the 226 nm photodissociation of jet-cooled O3. Consistent with previously reported trends, the rotational distribution is shifted to higher rotational states with decreasing wavelength. We observe highly suppressed odd rotational state populations due to a strong Λ-doublet propensity. The measured rotational distribution is in agreement with classical trajectory calculations for the v = 0 products, although the distribution is slightly narrower than predicted. The spatial anisotropy follows the previously observed trend of decreasing β with increasing photon energy with β = 0.72 ± 0.14 for v = 0, j = 38. As expected for a triatomic molecule, the v-j correlation is consistent with v perpendicular to j, but the measured correlation is nonlimiting due, in part, to rotational and translational depolarization. The j-dependent line width of the O2(a 1Δg) REMPI spectrum is also discussed in connection with the lifetime of the resonant O2(d 1Πg) state due to predissociation via the II 1Πg valence state.

Step-by-step state-selective tracking of fragmentation dynamics of water dications by momentum imaging

The double photoionization of a molecule by one photon ejects two electrons and typically creates an unstable dication. Observing the subsequent fragmentation products in coincidence can reveal a surprisingly detailed picture of the dynamics. Determining the time evolution and quantum mechanical states involved leads to deeper understanding of molecular dynamics. Here in a combined experimental and theoretical study, we unambiguously separate the sequential breakup via D+ + OD+ intermediates, from other processes leading to the same D+ + D+ + O final products of double ionization of water by a single photon. Moreover, we experimentally identify, separate, and follow step by step, two pathways involving the b 1Σ+ and a 1Δ electronic states of the intermediate OD+ ion. Our classical trajectory calculations on the relevant potential energy surfaces reproduce well the measured data and, combined with the experiment, enable the determination of the internal energy and angular momentum distribution of the OD+ intermediate.

Potential Nonstatistical Effects on the Unimolecular Decomposition of H2O2.

An attempt is made to evaluate the nonstatistical effects in the thermal decomposition of hydrogen peroxide (H2O2). Previous experimental studies on this reaction reported an unusual pressure dependence of the rate constant indicating broader falloff behavior than expected from conventional theory. In this work, the possibility that the rate constant is affected by nonstatistical effects is investigated based on classical trajectory calculations on the global potential energy surfaces of H2O2 and H2O2 + Ar. The emphasis is on the intramolecular energy redistribution from the K-rotor, that is, the external rotor for rotation around the principal axis of least moment of inertia. The calculations for the H2O2 molecules excited above the dissociation threshold suggest that the energy redistribution from the torsion and K-rotor to vibrations can be competitive with dissociation. In particular, the slow redistribution of the energy associated with the K-rotor significantly affects the dissociation rate. The successive trajectory calculations for collisions of H2O2 with Ar show that the energy associated with the K-rotor can be collisionally transferred more efficiently than the vibrational energy. On the basis of these results and several assumptions, a simple model is proposed to account for the nonstatistical effects on the pressure-dependent thermal rate constants. The model predicts significant broadening of the falloff curve of the rate constants but still cannot fully explain the experimental data.

A Hessian-Free Method to Prevent Zero-Point Energy Leakage in Classical Trajectories.

The problem associated with the zero-point energy (ZPE) leak in classical trajectory calculations is well known. Since ZPE is a manifestation of the quantum uncertainty principle, there are no restrictions on energy during the classical propagation of nuclei. This phenomenon can lead to unphysical results, such as forming products without the ZPE in the internal vibrational degrees of freedom (DOFs). The ZPE leakage also permits reactions below the quantum threshold for the reaction. We have developed a new Hessian-free method, inspired by the Lowe-Andersen thermostat model, to prevent energy dipping below a threshold in the local-pair (LP) vibrational DOFs. The idea is to pump the leaked energy to the corresponding local vibrational mode taken from the other vibrational DOFs. We have applied the new correction protocol on the ab-initio ground-state molecular dynamics simulation of the water dimer (H2O)2, which dissociates due to unphysical ZPE spilling from high-frequency OH modes. The LP-ZPE method has been able to prevent the ZPE spilling of the OH stretching modes by pumping back the leaked energy into the corresponding modes, while this energy is taken from the other modes of the dimer itself, keeping the system as a microcanonical ensemble.

Ab Initio Direct Dynamics.

The reactivity and dynamics of molecular systems can be explored computationally by classical trajectory calculations. The traditional approach involves fitting a functional form of a potential energy surface (PES) to the energies from a large number of electronic structure calculations and then integrating numerous trajectories on this fitted PES to model the molecular dynamics. The ever-decreasing cost of computing and continuing advances in computational chemistry software have made it possible to use electronic structure calculations directly in molecular dynamics simulations without first having to construct a fitted PES. In this "on-the-fly" approach, every time the energy and its derivatives are needed for the integration of the equations of motion, they are obtained directly from quantum chemical calculations. This approach started to become practical in the mid-1990s as a result of increased availability of inexpensive computer resources and improved computational chemistry software. The application of direct dynamics calculations has grown rapidly over the last 25 years and would require a lengthy review article. The present Account is limited to some of our contributions to methods development and various applications. To improve the efficiency of direct dynamics calculations, we developed a Hessian-based predictor-corrector algorithm for integrating classical trajectories. Hessian updating made this even more efficient. This approach was also used to improve algorithms for following the steepest descent reaction paths. For larger molecular systems, we developed an extended Lagrangian approach in which the electronic structure is propagated along with the molecular structure. Strong field chemistry is a rapidly growing area, and to improve the accuracy of molecular dynamics in intense laser fields, we included the time-varying electric field in a novel predictor-corrector trajectory integration algorithm. Since intense laser fields can excite and ionize molecules, we extended our studies to include electron dynamics. Specifically, we developed code for time-dependent configuration interaction electron dynamics to simulate strong field ionization by intense laser pulses. Our initial application of ab initio direct dynamics in 1994 was to CH2O → H2 + CO; the calculated vibrational distributions in the products were in very good agreement with experiment. In the intervening years, we have used direct dynamics to explore energy partitioning in various dissociation reactions, unimolecular dissociations yielding three fragments, reactions with branching after the transition state, nonstatistical dynamics of chemically activated molecules, dynamics of molecular fragmentation by intense infrared laser pulses, selective activation of specific dissociation channels by aligned intense infrared laser fields, angular dependence of strong field ionization, and simulation of sequential double ionization.

Two-Dimensional Master Equation Modeling of Some Multichannel Unimolecular Reactions.

Multichannel thermal decomposition reactions of n-propyl radicals, 1-pentyl radicals, and toluene are investigated by solving a two-dimensional master equation formulated as a function of total energy (E) and angular momentum (J). The primary aim of this study is to elucidate the role of angular momentum in the kinetics of multichannel unimolecular reactions. The collisional transition processes of the reactants colliding with argon are characterized based on the classical trajectory calculations and implemented in the master equation. The rate constants calculated by using the two-dimensional master equation are compared with those of one-dimensional master equations. The consequence of the explicit treatment of angular momentum depends on the J dependence of the microscopic rate constants and is particularly emphasized in the thermal decomposition of toluene, for which the C-H and C-C bond fission channels are considered. The centrifugal effect is insignificant in the energetically favored C-H bond fission but is substantial in the energetically higher C-C bond fission, which causes rotational channel switching of the microscopic rate constants. The proper treatment of the J-dependent channel coupling effect, weak collisional transfer of J, and initial-J-dependent collisional energy transfer are found to be essential for predicting the branching fractions at low pressures.

A detailed multiscale study of rotational–translational relaxation process of diatomic molecules

This article continues our cycle devoted to comprehensive investigation of the diatomic molecule collision process. In this paper, we focus particularly on the in-depth study of the rotational–translational (R–T) energy exchange process and Borgnakke–Larsen (BL) energy exchange model used in the direct simulation Monte Carlo method. The present study, which was performed on several levels of description (molecular, microscopic, and macroscopic), is based mainly on the highly detailed dataset (around 1011 configurations) of binary N2–N2 collisions, obtained via the classical trajectory calculation (CTC) method. This dataset, along with the explicit mathematical representation of the Borgnakke–Larsen model derived in the present paper, allowed us to obtain new results regarding the R–T energy exchange process: (1) we present an ab initio method to derive physically accurate expressions for inelastic collision probability pr in the BL model directly from CTC data; (2) we present a new two-parametric model for pr and compared it to the previously known models, including the recent nonequilibrium-direction-dependent model of Zhang et al. [“Nonequilibrium-direction-dependent rotational energy model for use in continuum and stochastic molecular simulation,” AIAA J. 52(3), 604 (2014)]; (3) it showed that apart from the well-known dependence of the rotational relaxation rate on “direction to equilibrium” (ratio between translational and rotational temperatures), on molecular scale, rotationally over-excited molecule pairs demonstrate almost zero energy transfer to the translational energy mode (even in the case of very significant discrepancies between translational and rotational energies); (4) it was also shown that the Borgnakke–Larsen approach itself may require reassessment since it fails to give a proper description of distribution of post-collision energies. Throughout this paper, we also tried to put together and analyze the existing works studying the rotational relaxation process and estimating the rotational collision number Zrot by performing reviews and assessment of (1) numerical approaches to simulate non-equilibrium problems, (2) models for inelastic collision probabilities pr, (3) approaches to estimate Zrot, and (4) intermolecular potentials used for molecular dynamics and CTC simulations. The corresponding conclusions are given in this paper.

Modeling third-body effects in the thermal decomposition of H2O2

The thermal decomposition of hydrogen peroxide (H2O2) in seven bath gases (M = He, Ar, H2, N2, CO, CH4, and H2O) has been studied by classical trajectory calculations of the collisional energy transfer processes and master equation analyses of the pressure-dependent rate constants. The energy transfer processes are modeled with the range parameter of the exponential down model and collision frequency for energy transfer. Both of the two quantities are calculated from the collisional trajectories propagated on the potential energy surfaces directly evaluated by the optimized spin-component-scaled MP2 method. The master equation calculations using these parameters were found to give reasonable descriptions of the rate constants at low pressures. The calculated relative third-body efficiencies agree well with the available experimental data for M = He, Ar, and N2 but the efficiency calculated for M = H2O appears to be overestimated at low temperature. The calculated rate constants are represented by the limiting high-pressure rate constant of k∞ = 6.7 × 1014 exp(−24800 K/T) s−1, limiting low-pressure rate constants for M = Ar and N2 of k0(Ar) = 3.65 × 108 (T/K)−4.691 exp(−26470 K/T) cm3 molecule−1 s−1 and k0(N2) = 8.21 × 109 (T/K)−5.034 exp(−26600 K/T) cm3 molecule−1 s−1, the center broadening factor of Fcent = 0.7 exp(− T/3400 K), and the tabulated relative third-body efficiencies. The pressure-dependent rate constants calculated for multicomponent bath gases are reasonably reproduced by the traditional linear mixture rule, whereas the mixture rule based on the reduced pressure is found to provide a more precise description.

Rotational relaxation model for nitrogen and its application in free jet expansion problem

This paper studies a rotational relaxation process in nitrogen. A series of more than 200 × 106 classical trajectory calculations (CTCs) of collisions of nitrogen molecules were carried out. Based on the CTC results, a collision model accounting for rotational relaxation was proposed and used for three-dimensional event-driven molecular dynamics simulation of a free jet expansion. It was shown that the collision model parameterized by the CTCs can produce results in good agreement with the existing experimental measurements.

Modeling Collisional Transitions in Thermal Unimolecular Reactions: Successive Trajectories and Two-Dimensional Master Equation for Trifluoromethane Decomposition in an Argon Bath.

Collisional transition processes in thermal unimolecular reactions are modeled by collision frequency, Z, and probability distribution function, P(E, J; E', J'), which describes the probabilities of collisional transitions from the initial state specified by the total energy and angular momentum, (E', J'), to the final states, (E, J). The validity of the collisional transition model, consisting of Z and P(E, J; E', J'), is assessed here for the title reaction. The present model and its parameters are derived from the moments of transition probabilities calculated by classical trajectory simulations. The model explicitly accounts for coupling between the energy and angular momentum transfer and the dependence of transition probability on the initial state. The performance of the model is evaluated by comparing the rate constants calculated by solving the two-dimensional master equation with those obtained from the classical trajectory calculations of the sequence of successive collisions. The rate constants are also compared with available experimental data. The present collisional transition model is found to perform fairly well for predicting the pressure-dependent rate constants. The uncertainty in the prediction and sensitivities of the rate constants to the model parameters are discussed. A simplified version of the model is proposed, which performs as well as the full model. The simplifications and robust procedures for calculating the model parameters are described.

Electron double-emission spectra for helium atoms in intense 400-nm laser pulses

Double photoelectron emission from He atoms by intense laser pulses with a wave length of $394.5\,nm$ is computed for intensities $3.5 - 9.2\times 10^{14}W/cm^2$. Joint momentum distributions confirm the characteristics seen in classical trajectory calculations. The pronounced transition from back-to-back to side-by-side emission with increasing intensity, the $He^{++}/He^+$ ratios, and a modulation of joint energy spectra agree well with a recent experiment [Henrichs et al., PRA 98, 43405 (2018)], if one admits an increase of experimental intensities by a factor $\sim 2$. We find that Freeman resonances enhance anti-correlated emission, we identify the signature of electron repulsion in joint angular distributions, and we interpret the modulation of joint energy spectra as a signature of multiple recollsions.

Direct Molecular Simulation of Nitrogen Dissociation Under Adiabatic Postshock Conditions

Direct molecular simulations (DMS) of nitrogen dissociation in adiabatic reservoirs covering a wide enthalpy range are presented. These conditions more realistically reproduce the gas state in a hypersonic shock layer than prior isothermal DMS studies. The Minnesota ab initio potential energy surface is used for nitrogen molecule-molecule () and nitrogen atom-molecule () classical trajectory calculations. Profiles of gas parameters (mixture composition, gas temperature, etc.) are reported, and the molecules’ internal energy population distributions during dissociation examined. It is observed that the evolution of the gas mixture under adiabatic conditions can be divided into two phases. Early on, excess translational energy is available for molecules to dissociate, regardless of their internal energy. This causes a sudden drop in the reservoir kinetic temperature and slows down subsequent dissociation. At later stages, the gas settles into a quasi-steady-state (QSS) regime, where dissociation proceeds primarily from higher-lying rovibrational levels near and above the dissociation threshold energy. The dissociation rate becomes limited by depletion of high-lying internal energy levels. Subsequently, adiabatic simulations are compared with prior isothermal DMS calculations performed at similar reservoir temperatures. The same shape is effectively observed for population distributions in both cases. This retroactively justifies the use of isothermal conditions to obtain QSS dissociation rate coefficients.

An atomic-level study of the N2–N2 collision process at temperatures up to 2000 K

This article studies the mechanics of the N2–N2 collision process at temperatures up to 2000 K through an extensive set of classical trajectory calculations of binary collisions. It is found that key postcollision characteristics, namely, the deflection angle and the rotational–translational energy exchange rate, are significantly affected by precollision values of the rotational energies of the molecules, which is not addressed in commonly used collision models. On the macroscopic scale, such a behavior will lead to viscosity collision cross section and relaxation rate becoming dependent on both translational and rotational temperatures, as well as on the form of the nonequilibrium rotational energy distribution.

Simplified Representation of Multichannel Thermal Unimolecular Reactions. II. Refined Parametrization of Formaldehyde Dissociation

Abstract Simplified representations of branching fractions for thermal unimolecular two-channel reactions are discussed. The dissociation of formaldehyde serves as an illustrative example. Quantum-corrected classical trajectory calculations on an ab initio potential energy surface are combined with master equation calculations for collisional energy transfer. The treatment accounts for roaming atom dynamics. The dependence of the channel branching fractions on the bath gas pressure and temperature, on the collision efficiencies, and on the difference of channel threshold energies, are explored. It is discussed to what extent the derived simplified representations of channel branching fractions can be generalized.

Profile of energy-loss spectrum in collisions of ions with heteronuclear diatomic molecules

SynopsisProfile of energy-loss spectrum in collisions of low velocity ions with heteronuclear diatomic molecules is theoretically investigated. With changing mass asymmetry in a target diatomic molecule, energy-loss spectrum is calculated with the classical trajectory calculation. Rotational excitation plays an essential role for energy-loss. The shape of the spectrum is found to show a drastic variation with mass asymmetry.

Energy-transfer in collisions of low velocity ions with CO molecules

SynopsisThe energy transfer in collisions of low velocity ions with CO molecules is theoretically investigated. Energy loss spectrum of ions scattered by CO molecules is obtained with using the classical trajectory (CT) calculation and the sudden-limit models. Both of the calculations well reproduce an experimental spectrum with a considerable accuracy. Rotational excitation is found to be principal energy transfer process from translational to internal motions.

Threshold photodissociation dynamics of NO2 studied by time-resolved cold target recoil ion momentum spectroscopy.

We study the near-threshold photodissociation dynamics of NO2 by a kinematically complete femtosecond pump-probe scheme using a cold target recoil ion momentum spectrometer. We excite NO2 to the optically bright Ã2B2 state with a 400 nm pulse and probe the ensuing dynamics via strong field single and double ionization with a 25 fs, 800 nm pulse. The pump spectrum spans the NO(X2Π) + O(3P) dissociation channel threshold, and therefore, following internal conversion, excited NO2 is energetically prepared both "above threshold" (dissociating) and "below threshold" (nondissociating). Experimentally, we can clearly discriminate a weak two-photon pump channel from the dominant single-photon data. In the single ionization channel, we observe NO+ fragments with nonzero momentum at 200 fs delay and an increasing yield of NO+ fragments with near-zero momentum at 3.0 ps delay. For double ionization events, we observe a time-varying Coulombic kinetic energy release between the NO+ and O+ fragments impulsively created from the evolving "hot" neutral ground state. Supported by classical trajectory calculations, we assign the decreasing Coulombic kinetic energy release at longer time delays to the increasing average NO-O distances in the ground electronic state during its large amplitude phase space evolution toward free products. The time-resolved kinetic energy release in the double ionization channel probes the large amplitude ground state evolution from a strongly coupled "inner region" to a loosely coupled "outer region" where one O atom is on average much further away from the NO. Both the time evolution of the kinetic energy release and the NO+ angular distributions support our assignments.