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Quantum oscillations in rotationally inelastic molecule-surface scattering: Energy dependence of transition probabilities
The Journal of Chemical Physics
Abstract
Exact quantum mechanical and quasiclassical trajectory results for rotationally inelastic N2 ‐corrugated surface collisions are compared over the energy range 0.01–0.04 eV. It is found that the degeneracy averaged, diffraction summed, rotationally inelastic transition probabilities display quantum oscillations.
The dynamical Lie algebraic (DLA) method is applied to statistical dynamics of energy transfer in rotationally inelastic molecule–surface scattering of NO molecules from Ag(111) surfaces. The statistical average values of the translational-to-rotational energy transfer and their dependence on main dynamical variables for the system, especially collision time, are obtained by the method in terms of the density operator formalism in statistical mechanics. It is shown that the DLA method appears to provide an alternative efficient technique to treat the energy transfer in the gas–surface scattering.
A dynamical Lie algebraic approach to statistical dynamics of the rotationally inelastic gas-surface scattering is described. This method is applied to the study of the scattering of NO from Ag(111) surface. Statistical average values of some physical observables, such as the translational-to-rotational (T→R) energy transfer and the interaction potential, and their dependence on various dynamic variables of the system are given analytically. The calculations predict a strong dependence of the average energy transfer and average interaction potential on temperature and the incident translational energy. The results imply that the dynamical Lie algebraic method appears to have a wide range of validity for describing the statistical dynamics of gas-surface scattering.
A computer program for calculating transition probabilities for rotationally and translationally inelastic scattering of homonuclear diatomic molecules from static, corrugated surfaces is described. The program uses an exact quantum-mechanical method which combines a close-coupling expansion of the internal states with a time-dependent wave-packet description of the center-of-mass motion. The wave function is propagated in time using an expansion of the time-evolution operator in a series of Chebyshev polynomials.
In this paper we generalize earlier work on potential scattering to atom–rigid rotor scattering. We compare six approaches including the interaction picture, modified Cayley, amplitude density, and symmetric split operator methods. All methods derive from the integral equation form of the time-dependent Schro¨dinger equation. The methods were tested using the standard Lester–Bernstein model potential. All methods were found to perform well with the same parameters. Fast Fourier transforms were not used in these methods, and an average execution time for a 16 channel problem on CRAY YMP supercomputer was about 45 s. This single calculation yields results at any energy significantly contained in the initial packet. In the present study, the S matrix was computed at a total of 42 energies, but results could have been obtained at many more energies without a large increase in computing time. Timing results for one of the methods are reported for 25, 64, 144, and 256 coupled channels.
The time-dependent Lippmann–Schwinger equation describing atom–diatom collisions is expressed in terms of a general reference Hamiltonian, Hr, whose dynamics are easily solved in one representation, and a corresponding disturbance Hamiltonian, Hd, whose dynamics are easily solved in a different representation. The wavefunction at time t + τ t is then expressed in terms of its value at a previous time t by means of a simple quadrature approximation. The resulting expression for ψ(t + τ) has a form similar to that occurring in earlier numerical unitary solutions to the time-dependent Schrodinger equation via a Cayley transformation. The structure of the new equations is made explicit for (a) the choice where Hr is taken to be the kinetic energy and Hd is the potential energy and (b) the choice where Hr is taken to be the potential energy and Hd is the kinetic energy. In addition, we also deal with several alternatives for treating the binding potential of the diatom. Several alternatives for choosing representations are then explored for reducing the equations to a form amenable to computation. The short time structure of the equations is discussed in terms of a multiple time-scales analysis. Keywords: molecular collisions, multiple time scales, quantum dynamics.
The close coupling wave packet (CCWP) and quasiclassical trajectory methods are used to study rotationally inelastic scattering of N2 from static, corrugated surfaces. The collision energy in these calculations ranges from 10 to 100 meV; 18 711 quantum states are included in the highest energy calculations to ensure convergence. The scattered molecules are analyzed with respect to the polarization of the final angular momentum vector and the amount of energy transferred into rotational motion and translational motion parallel to the surface. Comparisons of quantum and quasiclassical results show that quantum effects are important even with the relatively large mass of N2 and the high scattering energies used and can be seen even after summing over many final quantum states. A test of a factorization relation derived from the coordinate‐representation sudden (CRS) approximation gives qualitative agreement with the exact quantum results.
The dynamical Lie algebraic method is used for the description of statistical mechanics of rotationally inelastic molecule–surface scattering. A main advantage of this method is that it can not only give the expression for evolution operator in terms of the group parameters, but also provide the expression for the density operator for a given system. The group parameters may then be determined by solving a set of coupled nonlinear differential equations. Thus, the expressions of the statistical average values of the translational-to-rotational energy transfer, the interaction potential, and their dependence on the main dynamic variables for the system are derived in terms of the density operator formalism in statistical mechanics. The method is applied to the scattering of NO molecules from a static, flat Ag(111) surface to illustrate its general procedure. The results demonstrate that the dynamical Lie algebraic method can be useful for describing statistical dynamics of gas–surface scattering. © 2000 American Institute of Physics.
In this article we report an application of the time‐dependent close‐coupled wave‐packet (CCWP) method to the rotationally inelastic scattering of NO(X 2Π) molecules from a rigid, flat Ag(111) surface. Previous applications of the CCWP method have been restricted to either direct scattering off purely repulsive potentials requiring short propagation times, or weakly physisorbed systems in which only a few internal states are coupled. The calculations reported here were performed for a molecule in an initial state with a momentum distribution peaked around E=6700 cm−1 scattering off a strongly anisotropic potential with a well depth of 4400 cm−1 and a long‐range tail. Numerical procedures were introduced which enhance the efficiency of the CCWP method whenever a large number of internal states or a large number of grid points are needed to simulate the collision. For the current application to NO–Ag these techniques reduced the required CPU time by more than an order of magnitude. The resulting state–to–state transition probabilities are compared with previous time‐independent close‐coupled calculations, and with the semiclassical self‐consistent eikonal method (SCEM). The agreement between the two quantum‐mechanical methods is well within the accuracy of both numerical procedures. A comparison of the instantaneous transition probabilities calculated throughout the propagation shows good agreement between the CCWP and the SCEM calculations at high collision energies.
We present a detailed comparison of the efficiency and accuracy of the second-and third-order split operator methods, a time dependent modified Cayley method, and the Chebychev polynomial expansion method for solving the time dependent Schrodinger equation in the one-dimensional double well potential energy function. We also examine the efficiency and accuracy of the split operator and modified Cayley methods for the imaginary time propagation.
A new procedure for developing a coarse-grained representation of the free particle propagator in Cartesian, cylindrical, and spherical polar coordinates is presented. The approach departs from a standard basis representation of the propagator and the state function to which it is applied. Instead, distributed approximating functions (DAFs), developed recently in the context of propagating wave packets in 1-D on an infinite line, are used to create a coarse-grained, highly banded matrix which produces arbitrarily accurate results for the free propagation of wave packets. The new DAF formalism can be used with nonuniform grid spacings. The banded, discretized matrix DAF representation of <x\exp(-iK-tau/h)\x'> can be employed in any wave packet propagation scheme which makes use of the free propagator. A major feature of the DAF expression for the effective free propagator is that the modulus of the x(j),x(j), element is proportional to the Gaussian exp(-sigma(2)(0)(x(j) - x(f)2/2(sigma(4)(0) + h2-tau(2)/m2)). The occurrence of a tau-dependent width is a manifestation of the fundamental spreading of a wave packet as it evolves through time, and it is the minimum possible because the DAF representation of the free propagator is based on evolving the Gaussian generator of the Hermite polynomials. This suggests that the DAFs yield the most highly banded effective free propagator possible. The second major feature of the DAF representation of the free propagator is that it can be used for real time dynamics based on Feynman path integrals. This holds the possibility that the real time dynamics for multidimensional systems could be done by Monte Carlo methods with a Gaussian as the importance sampling function.
The wave packet that represents a physical molecular scattering system of interest evolves according to the time-dependent Schroedinger equation (TDSE), which is a linear, first order (in time), differential equation. In recent years, there has been considerable interest in the development and application of numerical methods for solving dynamics problems using time-dependent methods. A procedure for obtaining an approximate sparse (banded) matrix for the coordination representation of the free-particle propagator is presented. The technique takes advantage of the fact that the action of the free propagator on Hermite polynomials can be obtained analytically and represents the propagator in terms of its effect on the Hermite basis.
The rotationally inelastic molecule–surface scattering is analyzed using dynamical Lie algebraic method. We treat, by example, the simple model of the scattering of NO from a rigid, flat Ag(111) surface. The explicit expressions of transition probability and the probability current density are obtained. It is proved that dynamical Lie algebraic method can be useful for describing the scattering problems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 500–510, 2000
The dynamical Lie algebraic (DLA) method is used to describe statistical mechanics of energy transfer in rotationally inelastic molecule–surface scattering. Statistical average values of an observable for the scattering system are calculated in terms of density operator formalism in statistical mechanics. Employing a cubic expansion procedure of molecule–surface interaction potential leads to generation of a dynamical Lie algebra. Thus these statistical average values as a function of the group parameters can be obtained analytically in this formulation. The group parameters can be found from solving a set of coupled nonlinear differential equations. The DLA method, which has no need for determination of transition probabilities in advance as made routinely in the calculation, offers an efficient alternative to the method for computing the statistical average values. This method is much less computationally intensive because most of calculations can be analytically carried out. The average final rotational energies and their dependence on the main dynamic variables and the average interaction potential are presented for the rotationally inelastic scattering of NO molecules from a flat, static Ag(111) surface. Direct comparison is made between the predictions of this model calculation and experiment. The model reproduces well the degree of rotational excitation and correlation between the average final translational and the average rotational energies. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
This paper describes a dynamical Lie algebraic method that we have developed in the application of the theory of Alhassid and Levine [Phys. Rev. A 18 (1978) 89] to rotationally inelastic molecule–surface scattering. Transition probabilities and their dependence on main dynamics variables of the collision system can be given analytically. An application of the method to direct rotationally inelastic scattering of NO molecules from a static, flat Ag(111) surface is made. Calculations performed for this model system yield snapshots of the probability current density and those of the rate of change of probability density that provide an insight into the intimate details of the scattering dynamics in time. The results show that this method is efficient and more useful to the inelastic scattering problems.
The time-dependent form of the Lippmann-Schwinger integral equation is used as the basis of several new wave packet propagation schemes. These can be formulated in terms of either the time-dependent wave function or a time-dependent amplitude density. The latter is nonzero only in the region of configuratiaon space for which the potential is nonzero, thereby in principle obviating the necessity of large grids or the use of complex absorbing potentials when resonances cause long collision times (leading, consequently, to long propagation times). Transition amplitudes are obtained in terms of Fourier transforms of the amplitude density from the time to the energy domain. The approach is illustrated by an application to a standard potential scattering model problem where, as in previous studies, the action of the kinetic energy operator is evaluated by fast Fourier transform (FFT) techniques.
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
A numerically exact spectral method for solving the time-dependent Schroedinger equation in spherical coordinates is described. The angular dependence of the wave function is represented on a two-dimensional grid of evenly spaced points. The fast Fourier transform algorithm is used to transform between the angle space representation of the wave function and its conjugate representation in momentum space. The time propagation of the wave function is evaluated using an expansion of the time evolution operator as a series of Chebyshev polynomials. Calculations performed for a model system representing H2 scattering from a rectangular corrugated surface yield transition probabilities that are in excellent agreement with those obtained using the close-coupling wave packet (CCWP) method. However, the new method is found to require substantially more computation time than the CCWP method because of the large number of grid points needed to represent the angular dependence of the wave function and the variation in the number of terms required in the Chebyshev representation of the time evolution operator.
In this paper we describe the theory and application of the recently developed close‐coupling wave packet (CCWP) method to the study of transition probabilities of H2 scattered from flat and corrugated surfaces. We present an improved method of analyzing the final wave function which permits S matrices and transition probabilities to be obtained over a wide range of energies from the propagation of a single wave packet. Transition probabilities obtained using the CCWP method are in excellent agreement with those obtained from time‐independent close‐coupling (CC) calculations. For the present three‐dimensional H2‐corrugated surface scattering study, the CCWP calculations require roughly one‐tenth of the computation time of the CC method. Our results indicate that the CCWP method should be a very efficient method of obtaining highly accurate scattering results both for standard collision problems and for collision problems which cannot be readily treated using standard CC methods.
In this paper, we present the first formal and computational studies of Δmj transitions occurring in homonuclear molecule‐corrugated surface collisions. The model potential is a pairwise additive one which correctly incorporates the fact that Δmj transitions occur only for corrugated surfaces (provided the quantization axis is chosen to be the average surface normal). The principal results are: (a) Δmj transitions are extremely sensitive to lattice symmetry; (b) strong selection rules obtain for specular scattering; (c) the magnitude of Δmj ‐transition probabilities are strongly sensitive to surface corrugation; (d) the Δmj transitions depend strongly on diffraction peak; (e) the ratio of molecular length to lattice dimension (r/a) has a strong influence on the magnitude of Δmj ‐transition probabilities [with the probabilities increasing as (r/a) increases]; (f) Δmj rainbows are predicted to occur as a function of the (r/a) ratio increases; (g) Δmj transitions and the Δmj rainbow are expected to accompany Δj‐rotational rainbows; (h) such magnetic transition rainbows accompanying Δj rainbows are suggested as an explanation of recent experimental observations of quenching of NO polarization for larger Δj transitions in NO/Ag(111) scattering.
Numerically exact quantum calculations for scattering of N2 off a model corrugated rigid lattice are reported. The computations were done using the recently developed close coupling-wave packet method for treating quantum scattering. Results for all energies in the range 0.010–0.040 eV are obtained from a single wave packet propagation. The calculations included even N2 rotor j states j=0–12, all mj’s, and sufficient grid points to describe diffraction states −8≤m≤8, −4≤n≤4, for a total of 13 923 channels. The computations, which required a total of 230 minutes of computer time, were carried out on the CRAY2 supercomputer at the University of Minnesota Supercomputer Center under National Science Foundation support.
Two simple hard-cube models of gas-surface collisions are re-examined in light of recent argon-tungsten atomic beam scattering experiments. Both models provide a good description of the average energy exchange. The inclusion of a square well attraction to the hard-wall potential results in an accurate two parameter fit to the data. The derived well depth is in agreement with previous measurements of the heat of desorption.
Classical trajectory and quantum-mechanical calculations of final rotational state distributions are described for a simple model of the scattering of NO from Ag(111) which show strong effects of rotational rainbows. The quantum-mechanical calculation reproduces the general trends of the experimental data quite well. The classical calculation is qualitatively and quantitatively inadequate, an inadequacy which is remarkable in view of the large masses of the particles and the high degree of rotational excitation.
Empirical potential energy surfaces have been constructed to describe the nondissociative interaction of NO with the (111) faces of Ag and Pt. Stochastic trajectory simulations employing these interaction potentials accurately reproduce experimental angular and velocity scattering distributions. Measured rotational energy distributions of scattered molecules, including the observed ‘‘rotational rainbow’’ features, are also reproduced quantitatively. Arrhenius prefactors for desorption are computed to be large (1016 s−1), and the translational and rotational ‘‘temperatures’’ of desorbed molecules are found to be lower than the surface temperature, in agreement with experiment. Sticking probabilities, desorption rates, and the rotational energy of desorbed and scattered molecules are all found to be strongly influenced by the dependence of the attractive region of the gas‐surface potential on molecular orientation.
A new quantum mechanical time dependent integrator was used in the study of wave packet dynamics on potentials which include a deep well. The purpose of the study was to find the conditions, if any, for complex formation. The integrator, which is stable, conserves energy and norm and was used on the H++H2 system whose classical trajectory had been previously worked out. Almost no complex formation is found for the H++H2 system and its isotopic analogs. For high translational energies there was a good correspondence with the classical trajectory results, while for low translational energies where the classical trajectories become complex, the quantum calculations still show nonstatistical behavior. For even lower energies, a quantum effect took place leading to zero reactivity.
We have used the classical trajectory method to investigate rotationally inelastic encounters between diatomic molecules and a hierarchy of model surfaces: a rigid surface (RS), a simple harmonic oscillator (SHO), and a generalized Langevin oscillator (GLO). The diatom masses correspond to NO throughout, and the gas–surface interaction potential was invariant, with an attractive potential of ε=0.58 or 0.2 eV. Collision energies were 0.3 or 0.7 eV. Encounters were classified as ‘‘direct,’’ ‘‘indirect,’’ and ‘‘adsorbed.’’ Change from the RS to the SHO surface markedly increased the percentage of indirect encounters; change from SHO to GLO introduced adsorbed trajectories. Rainbow structure in the product rotational distribution, clearly evident on the RS, was obscured by the surface motion in the SHO and GLO models, remaining evident nonetheless for the higher collision energy. Sticking on the GLO surface decreased with increasing initial rotation, particularly for the weaker attractive potential. Consequently application of time‐reversal symmetry led to a yield of desorbing molecules weighted toward lower final rotation, i.e., to a rotational temperature TROT<TS (TS=surface temperature). The need for detailed experimental studies to establish the dynamics is evidenced by the fact that the characteristic bimodal distribution over final rotational states is obtained for all three model surfaces—RS, SHO, and GLO.
Rotational and reorientational transitions in molecular collisions with solid surfaces are investigated by a model based on a sudden approximation with respect to both the rotational and the diffraction states that play a role in the scattering. The approximation developed leads to computationally simple expressions and provides detailed insight into the physical properties of the processes involved. A detailed quantitative study is made of the rotational state distribution produced by the collision, the variation of rotational excitation probabilities with the scattering angle, and related questions. A number of factorizations, sum‐rule, and scaling properties are predicted for ‖ Sjmj,00;j′m′j′,mn @qL ‖2, the transition probability between the initial (jmj) and the final (j′m′j′) rotational states for scattering into the (mn) diffraction channel. The strongest sum rules and scaling laws are obtained using additional approximations beyond the sudden decoupling. Among the latter results: (1) The j,j′ dependence of ‖ Sj0,00;j’m’0,mn ‖2 is determined entirely by the difference variable Δj=j′−j. (2) The diffractive intensity distribution summed over all final rotational states is the same as that obtained for a mass‐equivalent atom (with an interaction that is the orientation‐averaged molecule–surface potential). (3) The rotational state distribution, summed over all diffraction states, equals that calculated from a corresponding flat surface. (4) All rotational transition probabilities for the (m,n) diffraction spot can be obtained from the diffraction–rotational transition probabilities in the (m,0) and (n,0) diffraction spots. The above and other properties are tested numerically in the framework of the full sudden approximation for a model of H2/LiF(001) in the energy range 0.5–0.9 eV. They are found to hold to excellent accuracy. Systematics of the results with regard to variation of the surface corrugation parameter are noted.
An approximate quantum mechanical approach to the scattering of systems with several degrees of freedom is developed. It is based on the exact (numerical) solution of the close coupled equations for subsets of the internal states and the coupling of these subsets to each other by perturbation theory. The calculation of the multichannel distorted wave solutions and their perturbative coupling is done simultaneously. Application to diffractive and rotationally inelastic scattering of H2 from a model corrugated surface is given and compared to earlier approximate diffractive sudden close coupled rotation calculations.
The lifetimes of rotationally-mediated resonances in the scattering of HD from a flat Ag surface are calculated by the quantum-mechanical complex rotation method and by classical trajectories. Qualitatively, considerable similarity is found between the quantum and the classical descriptions of the nature of the trapped states and of the mechanism of their decay. Quantitatively, quantum lifetimes are shorter than the corresponding classical values by as much as a factor of 4 to 7. Several interesting properties of the resonance states and trends associated with the lifetimes are observed, and simple physical interpretations offered.
Comparisons are made between calculations using the multi-channel distorted wave Born approximation (MDW), the diffractive sudden—close coupled rotation model (DSCCR), and the fully close coupled technique (CC) for a simple rigid rotor—rigid surface cohort Two different exponential repulsive potenials are considered and the surface corrugation is varied to display both perturbative and non-perturbative behavior. Corrugational effects are observed to be most pronounced for the less impulsive surface, as are departures from a strictly perturbative regime. Each approximation gives roughly comparable accuracy outside the specular peak; the DSCCR can be significantly more accurate than the MDW within the specular peak at higher corrugations.
A new method is presented for the solution of the time dependent Schrödinger equation in its application to physical and chemical molecular phenomena. The method is based on discretizing space and time on a grid, and using the Fourier method to produce both spatial derivatives, and second order differencing for time derivatives. The method conserves norm and energy, and preserves quantum mechanical commutation relations. One- and two-dimensional examples, where a comparison to analytic results is possible, are investigated.
The first computations suggesting the occurrence of rainbow scattering in the magnetic transitions occurring in molecule-corrugated surface scatteringare presented. The system studied models a Cl2 molecule scattered by a corrugated surface for which the potential is taken to be of the dumbbell type suggested by Gerber, Beard and Kouri (a sum of atom-Surface interactions). The magnetic transition probabilities which likely signal the Δmj rainbow are much larger than those observed in the only previous study of magnetic transitions in molecule-surface scattering
A new propagation scheme for the time dependent Schrödinger equation is based on a Chebychev polynomial expansion of the evolution operator Û=exp(−i Ĥt). Combined with the Fourier method for calculating the Hamiltonian operation the scheme is not only extremely accurate but is up to six times more efficient than the presently used second order differencing propagation scheme.