Classical Trajectory Method Research Articles

Coulomb-induced emission time shifts in high-order harmonic generation from H2+

Accurate emission times of high-order harmonic generation (HHG) are vital for high-precision ultrafast detection in attosecond science, but a quantitative analysis of Coulomb effects on this time is absent in the molecular HHG. Here, we investigate the Coulomb-induced emission-time shift in HHG of H2+ with two different internuclear distances R, where the times obtained via the Gabor transform of numerical data from solving the time-dependent Schrödinger equation are used as simulation experiment results. Based on the molecular strong-field approximation, we develop a trajectory-resolved classical model that takes into account the molecular two-center structure. By selecting appropriate electron trajectories and including Coulomb interactions, the classical trajectory method can reproduce Gabor emission times well. This consistence reveals that Coulomb tails cause an emission-time shift of ∼35 as at the R = 2.0 a.u. case and of ∼40-60 as at the R = 2.6 a.u. case under the present laser parameters when compared to the Coulomb-free quantum-orbit model. Our results are of significance to probe the attosecond dynamics via two-center interference.

Ion-atom-atom three-body recombination: From the cold to the thermal regime.

We present a study on ion-atom-atom reaction A + A + B+ in a wide range of systems and collision energies ranging from 100 μK to 105 K, analyzing two possible products: molecules and molecular ions. The dynamics is performed via a direct three-body formalism based on a classical trajectory method in hyperspherical coordinates developed in Pérez-Ríos et al. [J. Chem. Phys. 140, 044307 (2014)]. Our chief finding is that the dissociation energy of the molecular ion product acts as a threshold energy, separating the low- and high-energy regimes. In the low-energy regime, the long-range tail of the three-body potential dictates the fate of the reaction and the main reaction product. On the contrary, in the high-energy regime, the short-range of atom-atom and atom-ion interaction potential dominate the dynamics, enhancing molecular formation.

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

A partially linearized spin-mapping approach for simulating nonlinear optical spectra.

We present a partially linearized method based on spin-mapping for computing both linear and nonlinear optical spectra. As observables are obtained from ensembles of classical trajectories, the approach can be applied to the large condensed-phase systems that undergo photosynthetic light-harvesting processes. In particular, the recently derived spin partially linearized density matrix method has been shown to exhibit superior accuracy in computing population dynamics compared to other related classical-trajectory methods. Such a method should also be ideally suited to describing the quantum coherences generated by interaction with light. We demonstrate that this is, indeed, the case by calculating the nonlinear optical response functions relevant for the pump-probe and 2D photon-echo spectra for a Frenkel biexciton model and the Fenna-Matthews-Olsen light-harvesting complex. One especially desirable feature of our approach is that the full spectrum can be decomposed into its constituent components associated with the various Liouville-space pathways, offering a greater insight beyond what can be directly obtained from experiments.

Broadening and shifting of the carbon monoxide rotational lines in a wide temperature range: calculations in the framework of the classical impact theory for CO-He

The classical impact theory of Gordon is used to calculate half-widths and shifts of spectral lines of the pure rotational band of 12C16O isotopologue broadened by He. Two rotational transitions are examined: J=0-> J=1 and J=1-> J=2 in the wide temperature range from 1.3 to 600 K. The main purpose of this work is the study of the validity limits of classical impact theory at low temperatures. Dynamical calculations were performed on the accurate CO-He ab initio potential energy surface. The results of calculations are in good agreement with experimental data with the exception of very low temperatures. The contributions of collisions of different types (elastic, inelastic, quasibound complexes) are clearly examined in the classical picture frame. It is shown that the mismatches between classical theory and measurements are caused by the too high contribution of elastic collisions into broadening and shift in the present variant of theoretical model. The idea in the spirit of the Weisskopf theory is applied to try to diminish this contribution. The classical results are also compared with the results of fully quantum close coupling calculations made with using four CO-He interaction potentials. The roots of discrepancies at low temperatures as well as the virtues and the shortcomings of a classical approach are discussed. Keywords: collisional line broadening and shift, intermolecular interactions, classical impact theory, classical trajectory method, quasibound complexes.

Effect of Reactant's Rotational Excitation on Stereodynamic Properties of C+Sh(V=0, J=0-40)→Cs+H Reaction Investigated with Quasi Classical Trajectory Method

Effect of Reactant's Rotational Excitation on Stereodynamic Properties of C+Sh(V=0, J=0-40)→Cs+H Reaction Investigated with Quasi Classical Trajectory Method

Collisions of antiprotons with excited positronium atoms

A detailed theoretical investigation of the fundamental three-body antiproton-positronium system is presented, utilizing both quantum convergent close coupling and classical trajectory methods. We evaluate scattering cross sections for all major processes, including charge transfer and breakup, over a wide range of positronium kinetic energies and states with principal quantum number ${n}_{\mathrm{Ps}}\ensuremath{\le}6$. The study has significantly extended previous work, and the direct quantum-classical comparison has revealed a systematic approach to obtaining cross sections for arbitrarily high ${n}_{\mathrm{Ps}}$. This is of major significance for prospective applications of collisions of excited positronium atoms in cold (anti-)atomic physics.

Classical threshold law for the formation of van der Waals molecules.

We study the role of pairwise long-range interactions in the formation of van der Waals molecules through direct three-body recombination processes A + B + B → AB + B, based on a classical trajectory method in hyperspherical coordinates developed in our earlier works [J. Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014); M. Mirahmadi and J. Pérez-Ríos, J. Chem. Phys. 154, 034305 (2021)]. In particular, we find the effective long-range potential in hyperspherical coordinates with an exact expression in terms of dispersion coefficients of pairwise potentials. Exploiting this relation, we derive a classical threshold law for the total cross section and the three-body recombination rate yielding an analytical expression for the three-body recombination rate as a function of the pairwise long-range coefficients of the involved partners.

Including tunneling into the classical cross sections and rate constants for the N(2D) + H2 (v = 0, j = 0) reaction

The dynamics of the N(2D) + H2(v = 0, j = 0) reaction has been theoretically studied between 200 and 300 K by combining the classical trajectory method and the WKB penetration factor through the early barrier involved in this reaction. The resulting opacity functions, cross sections and rate constants are in reasonable agreement with accurate quantum wave packet results [S. Lin and H. Guo, J. Chem. Phys. 124, 031,101 (2006)] and experimental measurements [T. Suzuki et al., J. Chem. Soc., Faraday Trans, 1993, 89, 995]. Nevertheless, our treatment tends to underestimate the importance of deep tunneling, probably due to the crude linear approximation of tunneling paths employed in our study.

Extended virtual detector theory including quantum interferences

We extend an earlier “virtual detector” method [X. Wang, J. Tian, and J. H. Eberly, Phys. Rev. Lett. 110, 243001 (2013)], a hybrid quantum mechanical and classical trajectory method based on the concept of probability current, to include phases in the classical trajectories. Effects of quantum interferences, lost in the earlier method, are partially restored. The obtained photoelectron momentum distributions agree well with the corresponding numerical solutions of the time-dependent Schrödinger equation.

On the formation of van der Waals complexes through three-body recombination.

In this work, we show that van der Waals molecules X-RG (where RG is the rare gas atom) may be created through direct three-body recombination collisions, i.e., X + RG + RG → X-RG + RG. In particular, the three-body recombination rate at temperatures relevant for buffer gas cell experiments is calculated via a classical trajectory method in hyperspherical coordinates [Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014)]. As a result, it is found that the formation of van der Waals molecules in buffer gas cells (1 K ≲ T ≲ 10 K) is dominated by the long-range tail (distances larger than the LeRoy radius) of the X-RG interaction. For higher temperatures, the short-range region of the potential becomes more significant. Moreover, we notice that the rate of formation of van der Walls molecules is of the same order of the magnitude independent of the chemical properties of X. As a consequence, almost any X-RG molecule may be created and observed in a buffer gas cell under proper conditions.

When classical trajectories get to quantum accuracy: II. The scattering of rotationally excited H2 on Pd(111).

The classical trajectory method in a quantum spirit assigns statistical weights to classical paths on the basis of two semiclassical corrections: Gaussian binning and the adiabaticity correction. This approach was recently applied to the heterogeneous gas-surface reaction between H2 in its internal ground state and Pd(111) surface e.g. [A. Rodríguez-Fernández et al., J. Phys. Chem. Lett., 2019, 10, 7629]. Its predictions of the sticking and state-resolved reflection probabilities were found to be in surprisingly good agreement with those of exact quantum time-dependent calculations where standard quasi-classical trajectory calculations failed. We show in this work that the quality of the previous calculations is maintained or even improved when H2 is rotationally excited.

Continuum absorption of millimeter waves in nitrogen

We report the results of experimental study for the N2 absorption in the millimeter wavelength range at pressure and temperature ranging, respectively, from 750 to 1500 Torr and from 265 to 310 K. The obtained data are shown to be in a good agreement with the previously published experimental results, as well as with the predictions issued from available empirical models of nonresonant absorption and from the simulation of the N2–N2 rototranslational collision-induced absorption performed recently using the classical trajectories method relying on complete ab initio potential and induced dipole surfaces. Analysis of the first spectral moments leads to a conclusion that the role of true bound (N2)2 dimer absorption is negligible at least in the range of atmospheric temperatures. New experimental data reported in this paper are supported by our theoretical modelling being however somewhat in excess of the data presented in the HITRAN/CIA database for the same spectral range. We consider our data as an update of the previously available so-called “dry” atmospheric continuum which is conventionally employed nowadays in the mm-waves propagation models.

Effects of the Substituents on the Energy Flow of Toluene Derivatives in Collisions with N2 and O2

The effects of the substituents in the energy flow in four toluene derivatives (o‐chloro‐, o,‐bromo‐, α‐chloro‐, and α‐bromotoluene) on collision with N2 and O2 were investigated by classical trajectory methods. The energy loss by the α‐derivatives of toluene is significantly higher than that of the o‐derivatives. Furthermore, the substituent, Cl or Br, does not affect the energy loss. The dominant pathways for the vibrational decay of the excited o‐derivatives are the vibration–translation (V–T) and vibration–vibration (V–V) energy transfers to N2/O2. In contrast, the vibrational relaxation of the excited α‐derivatives occurs mainly through the V–T energy transfer. The efficiency of the intermolecular V–V energy transfer of the o‐derivatives is significantly higher than that of the α‐derivatives. In the o‐derivatives, as the excited energy (ET) increases, the excited C–H m bond frequency decreases; thus, the energy transfer via the V–V pathway increases. In the α‐derivatives, the V–V energy transfer is not efficient for both N2 and O2, although the C–H r and N2/O2 vibrations are near resonance.

Theoretical Study of Barrierless Chemical Reactions Involving Nearly Elastic Rebound: The Case of S(1D) + X2, X = H, D.

For some values of the total angular momentum consistent with reaction, the title processes involve nonreactive trajectories proceeding through a single rebound mechanism during which the internal motion of the reagent diatom is nearly unperturbed. When such paths are in a significant amount, the classical reaction probability is found to be markedly lower than the quantum mechanical one. This finding was recently attributed to an unusual quantum effect called diffraction-mediated trapping, and a semiclassical correction was proposed in order to take into account this effect in the classical trajectory method. In the present work, we apply the resulting approach to the calculation of opacity functions as well as total and state-resolved integral cross sections (ICSs) and compare the values obtained with exact quantum ones, most of which are new. As the title reactions proceed through a deep insertion well, mean potential statistical calculations are also presented. Seven values of the collision energy, ranging from 30 to 1127 K, are considered. Two remarkable facts stand out: (i) The corrected classical treatment strongly improves the accuracy of the opacity function as compared to the usual classical treatment. When the entrance transition state is tight, however, those trajectories crossing it with a bending vibrational energy below the zero point energy must be discarded. (ii) The quantum opacity function, particularly its cutoff, is finely reproduced by the statistical approach. Consequently, the total ICS is also very well described by the two previous approximate methods. These, however, do not predict state-resolved ICSs with the same accuracy, proving thereby that (i) one or several genuine quantum effects involved in the dynamics are missed by the corrected classical treatment and (ii) the dynamics are not fully statistical.

Strong-field ionization dynamics of asymmetric equilateral triatomic model molecules in circularly polarized laser fields

The strong-field-driven ionization dynamics of larger, nondiatomic molecules still constitutes a major challenge for a theoretical and numerical description. While a full numerical treatment of all electronic and nuclear degrees of freedom is computationally prohibitive, several approaches exist to model photoelectron spectra employing certain approximations. Among them are classical trajectory methods or the so-called strong-field approximation (SFA). Here we investigate in detail the strong-field ionization dynamics of simple, noncollinear triatomic model systems. For a detailed characterization of the ionization process of our system, we combine quantum dynamical simulations, modified classical trajectory calculations, and extensions of the SFA in order to analyze the complex pattern of the photoelectron momentum distribution. We aim at disentangling the contributions of excited states and the long-range character of the potential. We show that upon interaction with circularly polarized laser fields, the long-range character merely induces a small shift in the spectra, while the contribution of excited states is in several cases essential: in particular in near-infrared laser fields, compared to mid-infrared drivers, and for systems with larger internuclear distances, when excited electronic states are energetically closer. Key modifications are for our trajectory model to explicitly incorporate the numerically obtained tunnel barrier width directly into the ionization rate, thereby enabling a spatially anisotropic ionization rate, and an approximate treatment of the contribution of excited states via the Stark shift. In contrast, for the SFA, it is important to include electronically excited states for the description of the initial state. We also discuss the effect of the carrier-envelope phase on the ionization dynamics, as well as the influence of molecular averaging.

Simulation of experimental imaging results for four isotopic variants of the OH + CH4 reaction with a simple and relatively accurate theoretical approach

In 2005, Kopin Liu and co-workers used crossed molecular beam and velocity map imaging technics in order to measure pair-correlated product speed distributions for the OH + CH4 reaction and seven isotopic variants. We recently reproduced half of them with good accuracy by means of the classical trajectory method in a quantum spirit, which takes into account the semiclassical quantization of product vibrational motions. The second half are successfully reproduced here by using the same approach. The present results constitute one more proof of the ability of the classical trajectory method to provide satisfying predictions for polyatomic processes provided that a few quantum constraints are included in the calculations.

Energy Transfer and Bond Dissociation in the o‐chlorotoluene + H2/Cl2/HCl Collisions

Energy flow and C–H and C–Cl bond dissociations in o‐chlorotoluene (OCT) in collisions with H2, Cl2, and HCl are studied using classical trajectory methods. The energy loss by the excited OCT is small, but it increases with increasing total vibrational energy content ET of OCT between 5000 cm−1 and 30000 cm−1. The energy loss is larger in OCT + HCl collision than in OCT + H2 or OCT + Cl2. The difference is mainly due the near‐resonant conditions between the vibration of the incident molecule and the C–H or C–Cl vibrations. Intermolecular energy transfer occurs through vibration–translation (V–T) and vibration–vibration (V–V) pathways. The intramolecular energy flow between C–H and C–Cl in highly excited OCT (60000–60300 cm−1) leads to bond dissociation. The probabilities of C–H and C–Cl dissociation are 10−5–10−1 and increase exponentially with increasing vibrational excitation of OCT. The probabilities are found to increase when the incident molecule becomes heavier.

Isotopic Effects on Stereodynamics of the C+ + H2 → CH+ + H Reaction**Supported by the National Natural Science Foundation of China under Grant Nos. 11474141, 11274149, 11544015, the Program for Liaoning Excellent Talents in University under Grant No. LJQ2015040, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (2014-1685)

The effects of isotope substitution on stereodynamic properties for the reactions have been studied applying a quasi classical trajectory method occurring on the new ground state potential energy surface [J. Chem. Phys. 142 (2015) 124302]. In the center of mass coordinates applying the quasi classical trajectory method to investigate the orientation and the alignment of the product molecule. Differential cross section and three angle distribution functions P(θr), P(ϕr), P() on the potential energy surface that fixed the collision energy with a value is 40 kcal/mol have been studied. The isotope effect becomes more and more important with the reagent molecules H2 changing into HD and HT. P() as the joint probability density function of both polar angles θr and ϕr, which can illustrate more detailed dynamics information. The isotope effect is obvious influence on the properties of stereodynamics in the reactions of .

Effect of stable and metastable dimers on collision-induced rototranslational spectra: Carbon dioxide – rare gas mixtures

The role of stable and metastable dimers as well as of free collisions in the collision-induced rototranslational absorption by the compressed CO2–Ar and CO2–Xe gas mixtures is elucidated using the classical three-dimensional trajectories method. The contribution from the stable dimers is obtained via Fourier transform of the dipole correlation function. The spectral bandshape due to the unbound trajectories (metastable dimers and free collisions) is calculated as an averaged Fourier spectrum of the collision-induced dipole moment. The mean lifetimes of metastable dimers have been estimated as 3.8 ps for CO2–Ar and 5.9 ps for CO2–Xe pairs. Trajectory computations are complemented by calculations of zero spectral moments using pair distribution functions. The stable and metastable dimer contribution to the zero spectral moment is shown to be comparable with that from free collisions.