Classical Trajectory Method Research Articles

Nonadiabatic nuclear dynamics of atomic collisions based on branching classical trajectories

The branching classical trajectory method for inelastic atomic collision processes is proposed. The approach is based on two features: (i) branching of a classical trajectory in a nonadiabatic region and (ii) the nonadiabatic transition probability formulas particularly adapted for a classical trajectory treatment. In addition to transition probabilities and inelastic cross sections, the proposed approach allows one to calculate incoming and outgoing currents. The method is applied to inelastic Na + H collisions providing the results in reasonable agreement with full quantum calculations.

The kinetics study of the S + S2 → S3 reaction by the chaperone mechanism

The recombination of S atoms has been found to be stepwise from the smallest unit, the elemental S atom, to the most abundant molecule S(8). The reaction between S + S(2) → S(3) has not been reported either experimentally or by theory, but may be a key intermediate step in the formation of sulfur aerosols in low-O(2) atmospheres. In this work, the kinetics of this reaction is reported with Ar gas used as the chaperone molecule in the production of S(3) via two complex intermediates: SAr + S(2) and S(2)Ar + S. Quasi-classical and classical trajectory methods are used. The rate constant of the S + S(2) + Ar → S(3) + Ar reaction is determined to be 2.66 × 10(-33) cm(6) mol(-1) s(-1) at 298.15 K. The temperature dependence of the reaction is found to be 2.67 × 10(-33) exp[143.56(1∕T-1∕298.15)]. The second-order rate constant of S + S(2) → S(3) is 6.47 × 10(-14) cm(3) molecule(-1) s(-1) at 298.15 K and the Arrhenius-type rate constant is calculated to be 6.25 × 10(-14) exp[450.15(1∕T-1∕298.15)] cm(3) molecule(-1) s(-1). This work provides a rate coefficient for a key intermediate species in studies of sulfur formation in the modern Venus atmosphere and the primitive Earth atmosphere, for which assumed model rate coefficients have spanned nearly 4 orders of magnitude. Although a symmetry-induced mass-independent isotope effect is not expected for a chaperone mechanism, the present work is an important step toward evaluating whether mass-independence is expected for thiozone formation as is observed for ozone formation.

Quasi-Classical Trajectory Study on the Reaction Kinetics of Li+HF(<em>ν</em>=0, <em>j</em>=0)→LiF+H

A theoretical study of the Li+HF (ν=0, j=0)→LiF+H reaction was carried out using the quasi- classical trajectory (QCT) method based on the latest APW potential energy surface (PES) obtained by Aguado et al. The reaction cross-section, rotational alignment, and angular distributions of the products were obtained at different collision energies. The results indicate that there are two reaction pathways, i.e., an abstraction pathway and an insertion pathway for this reaction. At a low collision energy the insertion mechanism is dominant whereas at high energy (E>200 meV) the abstraction mechanism is dominant.

Ab initio intermolecular potential energy surface and thermophysical properties of hydrogen sulfide

A six-dimensional potential energy hypersurface (PES) for two interacting rigid hydrogen sulfide molecules was determined from high-level quantum-mechanical ab initio computations. A total of 4016 points for 405 different angular orientations of two molecules were calculated utilizing the counterpoise-corrected supermolecular approach at the CCSD(T) level of theory and extrapolating the calculated interaction energies to the complete basis set limit. An analytical site-site potential function with eleven sites per hydrogen sulfide molecule was fitted to the interaction energies. The PES has been validated by computing the second pressure virial coefficient, shear viscosity, thermal conductivity and comparing with the available experimental data. The calculated values of volume viscosity were not used to validate the potential as the low accuracy of the available data precluded such an approach. The second pressure virial coefficient was evaluated by means of the Takahashi and Imada approach, while the transport properties, in the dilute limit, were evaluated by utilizing the classical trajectory method. In general, the agreement with the primary experimental data is within the experimental error for temperatures higher than 300 K. For lower temperatures the lack of reliable data indicates that the values of the second pressure virial coefficient and of the transport properties calculated in this work are currently the most accurate estimates for the thermophysical properties of hydrogen sulfide.

Evolution of a pair of classical ions in a cavity with elastic walls, crosspieces, and implanted charges

Using the method of classical trajectories, we have simulated the motion of a pair of oppositely charged ions in spherical and ellipsoidal cavities, including cavities that contain cylindrical “crosspieces” and positively charged “nuclei.” We supposed that each of the ions reflected off the cavity wall, crosspieces, and nuclei according to the elastic impact law. Most of attention is focused on the statistics of the events of ion recombination and those of dissociation of the corresponding molecule with an ionic bond. When nuclei are absent, recombination and dissociation events are possible only on collisions of the ions with the cavity wall or crosspieces. In the presence of nuclei, on the other hand, the major part of the events occur in time intervals between the collisions of the ions with the cavity wall, crosspieces, or nuclei.

Comparative dynamical study of the HD and [formula omitted] isotopomers interacting with Pd(1 1 1) and Cu(1 1 0) surfaces

We study the dynamics of HD and H 2 molecules interacting with Pd(1 1 1) and Cu(1 1 0) using the classical trajectory method based on potential energy surfaces obtained from Density Functional Theory calculations. Our results predict a negligible isotopic effect on the dissociative adsorption probability on Pd(1 1 1) whereas on Cu(1 1 0), the adsorption probability for HD ( ν i = 0 ) is slightly lower than for H 2 ( ν i = 0 ) , mainly due to its lower initial vibrational zero point energy. The final rotational energy distribution of scattered HD and H 2 molecules are very similar. This shows that the asymmetric mass distribution of HD, barely affects the fraction of initial translational energy transferred to rotation during the scattering process. Our calculations point to the larger number of open rotational excitation channels for HD, as the main cause of rotational excitation probabilities larger than for H 2 . The theoretical apparent rotational temperature, T rot , of HD molecules scattered from Pd(1 1 1) at impact energy E i = 75 meV , is in good agreement with the experimental value. In contrast, for Cu(1 1 0) the theoretical T rot is much lower than the value measured for Cu(1 0 0). Possible reasons for such a discrepancy between theory and experiments are discussed.

Calculation of the transport and relaxation properties of dilute water vapor

Transport properties of dilute water vapor have been calculated in the rigid-rotor approximation using four different potential energy hypersurfaces and the classical-trajectory method. Results are reported for shear viscosity, self-diffusion, thermal conductivity, and volume viscosity in the dilute-gas limit for the temperature range of 250-2500 K. Of these four surfaces the CC-pol surface of Bukowski et al. [J. Chem. Phys. 128, 094314 (2008)] is in best accord with the available measurements. Very good agreement is found with the most accurate results for viscosity in the whole temperature range of the experiments. For thermal conductivity the deviations of the calculated values from the experimental data increase systematically with increasing temperature to around 5% at 1100 K. For both self-diffusion and volume viscosity, the much more limited number of available measurements are generally consistent with the calculated values, apart from the lower temperature isotopically labeled diffusion measurements.

Calculation of the transport and relaxation properties of methane. II. Thermal conductivity, thermomagnetic effects, volume viscosity, and nuclear-spin relaxation

Transport properties of pure methane have been calculated in the rigid-rotor approximation using the recently proposed intermolecular potential energy hypersurface [R. Hellmann et al., J. Chem. Phys. 128, 214303 (2008)] and the classical-trajectory method. Results are reported in the dilute-gas limit for the temperature range of 80-1500 K. The calculated thermal conductivity values are in very good agreement with the measured data and correlations. In the temperature range of 310-480 K the calculated values underestimate the best experimental data by 0.5%-1.0%. We suggest that the calculated values are more accurate, especially at low and high temperatures, than the currently available correlations based on the experimental data. Our results also agree well with measurements of thermal transpiration and of the thermomagnetic coefficients. We have shown that although the dominant contribution to the thermomagnetic coefficients comes from the Wjj polarization in the spherical approximation, the contribution of a second polarization, Wj, cannot be neglected nor can a full description of the Wjj polarization. The majority of the volume viscosity measurements around room temperature are consistent with the calculated values but this is not the case at high and low temperatures. However, for nuclear-spin relaxation the calculated values consistently exceed the measurements, which are mutually consistent within a few percent.

Calculation of charge-changing cross-sections of ions or atoms colliding with fast ions using the classical trajectory method

Evaluation of ion–atom charge-changing cross-sections is needed for many accelerator applications. A Classical Trajectory Monte Carlo (CTMC) simulation has been used to calculate ionization and charge-exchange cross-sections. For benchmarking purposes, an extensive study has been performed for the simple case of hydrogen and helium targets in collisions with various ions. Despite the fact that the simulation only accounts for classical mechanics, the calculations are comparable to experimental results for projectile velocities in the region corresponding to the vicinity of the maximum cross-section. The shortcomings of the CTMC method for multielectron target atoms are discussed.

Calculation of the transport and relaxation properties of methane. I. Shear viscosity, viscomagnetic effects, and self-diffusion

Transport properties of pure methane gas have been calculated in the rigid-rotor approximation using the recently proposed intermolecular potential energy hypersurface [R. Hellmann et al., J. Chem. Phys. 128, 214303 (2008)] and the classical-trajectory method. Results are reported in the dilute-gas limit for shear viscosity, viscomagnetic coefficients, and self-diffusion in the temperature range of 80-1500 K. Compared with the best measurements, the calculated viscosity values are about 0.5% too high at room temperature, although the temperature dependence of the calculated values is in very good agreement with experiment between 210 and 390 K. For the shear viscosity, the calculations indicate that the corrections in the second-order approximation and those due to the angular-momentum polarization are small, less than 0.7%, in the temperature range considered. The very good agreement of the calculated values with the experimental viscosity data suggests that the rigid-rotor approximation should be very reasonable for the three properties considered. In general, the agreement for the other measured properties is within the experimental error.

Classical calculation of self-broadening in N2 Raman spectra

This paper reports on a detailed study of self-broadening of Raman lines in gaseous N2 using classical trajectory method. Formulas of classical impact theory of spectral lines are used along with derived exact 3D Hamilton equations in body-fixed coordinates for N2–N2 binary collisions. Molecules are treated as rigid rotors, several available atom-atom + electrostatic intermolecular potentials are used. Q- and S-branch linewidths of pure N2 are computed for the rotational quantum numbers J = 0–30 at different temperatures ranging from 113 to 2400 K. The validity of mean thermal velocity approximation is tested. The results of calculations are compared with existing experimental data and with the predictions of semiclassical Robert–Bonamy method. The agreement between these classical results and experiment is found to be good, better than the results using the Robert–Bonamy method. Special consideration is given to temperature dependence of linewidths and its approximation by simple power law. The contributions of elastic and inelastic collisions to N2 self-broadening are analysed in the light of experimentally observed differences between Q- and S-branch linewidths. The simple and obvious classical method is found to be capable of accurate and rapid predictions of collisional line self-broadening of nitrogen molecules in binary collisions.

Strong field approximation for systems with Coulomb interaction

A theory describing above-threshold ionization of atoms and ions in a strong electromagnetic field is presented. It is based on the widely known strong field approximation and incorporates the Coulomb interaction between the photoelectron and the nucleus using the method of complex classical trajectories. A central result of the theory is the Coulomb-corrected ionization amplitude whose evaluation requires little extra numerical effort. By comparing our predictions with the results of ab initio numerical solutions for two examples we show that the new theory provides a significant improvement of the Coulomb-free strong field approximation. For the case of above-threshold ionization in elliptically polarized fields a comparison with available experimental data is also presented.

The sticking of H and D atoms on a graphite (0001) surface: The effects of coverage and energy dissipation

Classical trajectory methods are used to examine the trapping and sticking of H and D atoms on the graphite (0001) surface. Total energy calculations based on density functional theory are used to construct the model potential energy surface, and graphite clusters of up to 121 atoms are considered. For hydrogen to chemisorb, the bonding carbon must pucker out of the surface plane by roughly 0.4 A. For incident energies above the 0.2 eV barrier, any trapped H atoms must rapidly dissipate their excess energy into the surrounding lattice within a few vibrations of the C-H stretch in order to remain bound. For sufficiently large clusters, the C-H bond stabilizes within about 0.1 ps. The sticking probability for D at 150 K is in the range of 5%-10%, more-or-less consistent with the most recent measurements in the limit of zero coverge. Variation with isotope and substrate temperature is weak. We estimate that the sticking cross section for adsorption at the para site, directly across the sixfold carbon ring from an already adsorbed H atom, can be four or more times larger that the zero coverage sticking cross section.

A comparative multi-property analysis of existing models for the He–N2 potential energy surface

The ability of an improved version of a recent three-dimensional ab initio potential energy surface for the He–N2 interaction [Phys. Rev. A 66, 042703 (2002)], determined from high-level coupled-cluster calculations (including full singles and doubles, perturbative triples, and Brueckner orbitals), to predict scattering cross-sections and various bulk gas mixture properties is tested. The full three-dimensional potential energy surface has been employed for the calculation of vibrational relaxation rates, and a two-dimensional version (at a fixed N2 bond length of 2.0743 a 0) has been used for the calculation of molecular beam scattering cross-sections using quantum close-coupling methods and for the calculation of bulk gas phenomena using classical trajectory methods. The results obtained from the two-dimensional version of the present potential energy surface are compared with those obtained from three other recent accurate two-dimensional representations of the He–N2 interaction.

Classical dynamics of 3D Hydrogen molecular ion in intense laser fields

The classical trajectory method is used to study the dynamics of 3D Hydrogen molecular ion $${\rm H}_{2}^{+}$$ interacting with intense laser fields. In the 3D classical model, a three-body Hamiltonian with one-dimensional nuclear motion restricted to the direction of the laser field is considered. The motion of electron and nucleus is described by the classical Hamiltonian canonical equations. The probabilities of ionization, dissociation and Coulomb explosion as functions of time are calculated and the average distances from electron to the mass-center for various laser parameters are implemented by symplectic method. The dynamics of $${\rm H}_{2}^{+}$$ in two-color laser fields are also investigated. We compare our results with the corresponding quantum-mechanical calculations and find they produce similar qualitative features in many cases.

Anisotropic screening effects on elastic eikonal scattering cross section in magnetized plasmas

The anisotropic plasma screening effects on the eikonal scattering cross section for the electron–ion elastic collision are investigated in anisotropic plasmas in the presence of an externally applied magnetic field. The second-order eikonal analysis is employed to investigate the elastic collision process in magnetized plasmas. The classical straight line trajectory method is applied to the motion of the projectile electron in order to investigate the variation of the second-order eikonal phase as a function of the impact parameter and plasma parameters. The result shows that the major contribution of the magnetic field effect is found to be given by the second-order eikonal phase. It is also found that the eikonal scattering cross section including the anisotropic screening effect described by the effective interaction potential is always greater than that including the static screening effect obtained by the Debye–Hückel model. In addition, it is found that the eikonal cross section has the maximum value when the rotational energy with the Larmor frequency is equal to the thermal energy.

Observation of anisotropic interactions between metastable atoms and target molecules by two-dimensional collisional ionization electron spectroscopy

Excited metastable atoms colliding with target molecules can sensitively probe outer properties of molecules by chemi-ionization (Penning ionization) from molecular orbitals in the outer region, since metastable atoms cannot penetrate into the repulsive interaction wall around the molecules. By means of two-dimensional measurements using kinetic energy analysis of electrons combined with a velocity-resolved metastable beam, one can obtain information on the anisotropic interaction between the colliding particles without any control of orientation or alignment of target molecules. We have developed a classical trajectory method to calculate the collision energy dependence of partial ionization cross-sections (CEDPICS) on the anisotropic interaction potential energy surface, which has enabled us to study stereodynamics between metastable atoms and target molecules as well as the spatial distribution of molecular orbitals and electron ejection functions which have a relation with entrance and exit channels of the reaction. Based on the individual CEDPICS, the electronic structure of molecules can also be elucidated.

Associative desorption of N 2 from Ru(0 0 0 1): A computational study

We have studied the associative desorption of N 2 from a Ru(0 0 0 1) surface, using a six-dimensional classical trajectory method and a potential energy surface based on density functional theory. The frozen surface and Born–Oppenheimer approximations were used, neglecting phonons and electron–hole pair excitation. Our results show, in agreement with experiments, that only 40% of the potential energy at the barrier is transferred to the translational and rotational degrees of freedom. Our results also show, in disagreement with experiment, high vibrational excitation of the desorbed molecules, suggesting the importance of non-adiabatic effects for this process.

Theoretical investigation of the dissociation dynamics of vibrationally excited vinyl bromide on an ab initio potential-energy surface obtained using modified novelty sampling and feed-forward neural networks

The reaction dynamics of vibrationally excited vinyl bromide have been investigated using classical trajectory methods on a neural network potential surface that is fitted to an ab initio database of 12 122 configuration energies obtained from electronic structure calculations conducted at the MP4(SDQ) level of theory using a 6-31G(d,p) basis set for the carbon and hydrogen atoms and Huzinaga's (43334334) basis set augmented with split outer s and p orbitals (4332143214) and a polarization f orbital with an exponent of 0.5 for the bromine atom. The sampling of the 12-dimensional configuration hyperspace of vinyl bromide prior to execution of the electronic structure calculations is accomplished by combining novelty-sampling methods, chemical intuition, and trajectory sampling on empirical and neural network surfaces. The final potential is obtained using a two-layer feed-forward neural network comprising 38 and 1 neurons, respectively, with hyperbolic tangent sigmoid and linear transfer functions in the hidden and output layers, respectively. The fitting is accomplished using the Levenberg-Marquardt algorithm with early stopping and Bayesian regularization methods to avoid overfitting. The interpolated potentials have a standard deviation from the ab initio results of 0.0578 eV, which is within the range generally regarded as "chemical accuracy" for the purposes of electronic structure calculations. It is shown that the potential surface may be easily and conveniently transferred from one research group to another. The files required for transfer of the vinyl bromide surface can be obtained from the Electronic Physics Auxiliary Publication Service. Total dissociation rate coefficients for vinyl bromide are obtained at five different excitation energies between 4.50 and 6.44 eV. Branching ratios into each of the six open reaction channels are computed at 24 vibrational energies in the range between 4.00 and 6.44 eV. The distribution of vibrational energies in HBr formed via three-center dissociation from vinyl bromide is determined and compared with previous theoretical and experimental results. It is concluded that the combination of ab initio electronic structure calculations, novelty sampling with chemical intuition and trajectories on empirical analytic surfaces, and feed-forward neural networks provides a viable framework in which to execute purely ab initio molecular-dynamics studies on complex systems with multiple open reaction channels.

Transport properties of H–N2 mixtures

Transport coefficients (shear viscosity, volume viscosity, thermal conductivity, and mass and thermal diffusion coefficients) of H–N2 mixtures in the dilute-gas limit have been calculated from the intermolecular potential in the temperature range 300–2000K using the classical trajectory method. The intermediate results pertaining to H–N2 binary interactions are reported, mainly in terms of cross-section ratios. Cross-sections evaluated with the Mason–Monchick approximation yield very good results for this system, the largest deviations, about 2.5%, being observed for the thermal diffusion coefficient. The accuracy here of this approximation can primarily be attributed to a light H atom and a weakly non-spherical potential resulting in a high rotational collision number. Furthermore, we investigate to which H–N2 cross-sections and their ratios the values of the mixture transport coefficients are most sensitive. Our results indicate that, for some cross-section ratios, reliance on universal correlations at high temperatures, often used in flame codes, can induce sizeable errors in the thermal conductivity coefficient and especially in the thermal diffusion coefficients. We also observed that the volume viscosity is particularly sensitive to the value of the cross-section for internal energy relaxation in H–N2 collisions.