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Molecule–corrugated surface collisions: Converged close coupling wave packet and quasiclassical trajectory calculations for N2 scattering from corrugated lattices
The Journal of Chemical Physics
Abstract
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 N 2 /LiF(001) surface scattering problem has been investigated by the MC-TDH scheme [52], and by other methods [159,160], for scattering energies up to 100 meV. We recently returned to this problem and studied it over an extended energy range of 60 to 300 meV [107]. ...
... The recent technical advances, such as the correction scheme (see Sec. 7.2.2) and the use of CAPs (see Sec. 4.7) combined with the flux analysis (see Sec. 8.6), allowed the z-grid length to be shortened by roughly a factor of three. However, because of the higher energy the grid spacings must now be finer, making the problem much larger than the one studied previously [52,159,160]. A converged calculation requires (N x , N y , N z , N θφ ) = (24,24,108,223) grid points or primitive basis functions (j max = 36, m j,max = 6, and taking only even j's leads to 223 spherical harmonics as basis functions). ...
... The diffraction channel averaged transition probability to final rotational states depicts a strong rotational rainbow structure in particular for j f inal ≥ 10 and E ≥100 meV. More details on results may be found in [52,159] and in a forthcoming publication [107]. ...
A review is given on the multi-configuration time-dependent Hartree (MCTDH) method, which is an algorithm for propagating wavepackets. The formal derivation, numerical implementation, and performance of the method are detailed. As demonstrated by example applications, MCTDH may perform very eciently, especially when there are many (typically four to twelve, say) degrees of freedom. The largest system treated with MCTDH to date is the pyrazine molecule, where all 24 (!) vibrational modes were accounted for. The particular representation of the MCTDH wavefunction requires special techniques for generating an initial wavepacket and for analysing the propagated wavefunction. These techniques are discussed. The full efficiency of the MCTDH method is only realised if the Hamiltonian can be written as a sum of products of one-dimensional operators. The kinetic energy operator and many model potential functions already have this required structure. For other potential functions, we describe an e...
... Nousétudions la diffraction d'une molécule N 2 sur une surface de fluorure de Lithium LiF(001). Le modèle que nous avons utilisé aété introduit par Gerber et al. [112] et a depuiś eté largement utilisé par d'autres groupes [113,114,115,116]. Auxénergies de collision considérées, le premier niveau vibrationnel de la molécule est inaccessible ; N 2 est considéré comme un rotateur rigide. ...
... La surface est donc représentée par les variables X, et Y , parallèlesà la surface. Le potentiel d'intéraction est celui proposé par Gerber et al. [112], et qui a depuisété largement utilisé pour modéliser ce système [113,114,115,116]. Il s'écrit : Dans l'étude qui va suivre, nous n'avons considéré que desénergies "thermiques", inférieuresà 300 meV. ...
The subject of this mainly methodological thesis is the use of quantum trajectories, defined by de Broglie and Bohm, in studing molecular dynamical processes. Two kind of studies are presented. On the first hand, we use the quantum trajectories to solve the hydrodynamical form of the Schrödinger equation. A numerical method which combine the use of a fixed grid and moving grids was developed and applied to the photodissociation of the H2 molecule. This method is numerically efficient, especially for process like direct dissociation. On the second hand, we use quantum trajectories to establish a new hybrid quantum / classical propagation scheme. Methods of this kind are useful to treat the dynamics of systems too large to be treated by quantum mechanics, but where however at least some degrees of freedom require a quantum treatment. In our method, the positions associated with the quantum trajectories are used in the equation for the classical degrees of freedom to calculate their reaction to the quantum part. The results obtained on three systems, simple enough to have access to the exact results, are compared to those obtained by other hybrid schemes already widely used.
... However, knowledge of higher degrees of molecular alignment provides new information that cannot otherwise be obtained. In molecular beam scattering of diatomic molecules from crystal surfaces, the measurement of the hexadecapolar alignment of the rotationally excited products provides additional and complimentary information to quadrupolar alignment, [48][49][50] and in two-and multiphoton molecular photodissociation the utility of measuring the higher order moments of the photofragment angular momentum M J distributions has been recognized. 51,52 In condensed phases, molecular ordering with moments of rank K > 2 is found in a wide range of environments, including liquid crystals, lipid bilayers, cell membranes, and at interfaces. ...
Time and polarization-resolved stimulated emission depletion (STED) measurements are used to investigate excited state evolution following the two-photon excitation of enhanced green fluorescent protein (EGFP). We employ a new approach for the accurate STED measurement of the hitherto unmeasured degree of hexadecapolar transition dipole moment alignment α40 present at a given excitation-depletion (pump-dump) pulse separation. Time-resolved polarized fluorescence measurements as a function of pump-dump delay reveal the time evolution of α40 to be considerably more rapid than predicted for isotropic rotational diffusion in EGFP. Additional depolarization by homo-Förster resonance energy transfer is investigated for both α20 (quadrupolar) and α40 transition dipole alignments. These results point to the utility of higher order dipole correlation measurements in the investigation of resonance energy transfer processes.
... [5][6][7] Together, these methods enabled very large-scale calculations on inelastic and reactive scattering. [8][9][10][11] Although the wave packet method is very successful, a bottleneck may arise if the scattering is affected by the temporary population of metastable states ''resonances''. [12][13][14] This is obvious for a time-dependent TD formalism: If the resonances have a long lifetime, a long propagation time will be required for allowing the resonances to ''leak away'' to the asymptotic regions. ...
We investigate the usefulness of a hybrid method for scattering with resonances. Wave packet propagation is used to obtain the time‐dependent wave function Ψ(t) up to some time T at which direct scattering is over. Next, Ψ(t) is extrapolated beyond T employing resonance eigenvalues and eigenfunctions obtained in a Lanczos procedure, using Ψ(T) as starting vector to achieve faster convergence. The method is tested on one two‐dimensional (2D) and one four‐dimensional (4D) reactive scattering problem, affected by resonances of widths 0.1–5 meV. Compared to long time wave packet propagation, the hybrid method allows large reductions in the number of Hamiltonian operations NH required for obtaining converged reaction probabilities: A reduction factor of 24 was achieved for the 2D problem, and a factor of 6 for the 4D problem. © 1996 American Institute of Physics.
The first book dedicated to this new and powerful computational method begins with a comprehensive description of MCTDH and its theoretical background. There then follows a discussion of recent extensions of MCTDH, such as the treatment of identical particles, leading to the MCTDHF and MCTDHB methods for fermions and bosons. The third section presents a wide spectrum of very different applications to reflect the large diversity of problems that can be tackled by MCTDH. The result is handbook and ready reference for theoretical chemists, physicists, chemists, graduate students, lecturers and software producers.
The multiconfiguration time-dependent Hartree (MCTDH) method is applied to rotational and diffractive inelastic molecule-corrugated surface scattering. The molecule is treated as a rigid rotor, hence there are five degrees of freedom included in the calculation. The model systems H2/rectangular lattice and N2/LiF (001) are investigated for scattering with normal incidence. The performance and reliability of the MCTDH method is critically examined with respect to the structure of the MCTDH wave function and the choice of the basis set representation. The MCTDH reproduces the fine details of the state-to-state transition probabilities calculated by the numerically exact close-coupled wave packet (CCWP) method. We show that it is useful to represent two of the internal degrees of freedom by one set of single-particle functions when these degrees are strongly coupled, or when their MCTDH-contraction efficiency is low.
The H2+LiF(001) system was used to investigate the performance of the hybrid close-coupling wave packet (CCWP) method and of a symmetry adapted, fully close-coupled wave packet (SAWP) method for a molecule–surface problem characterized by fairly high corrugation. In the calculations, a realistic, ϕ-dependent model potential was used. The calculations were performed for a collision energy of 0.2 eV, with H2 initially in its j=0 rotational state at normal incidence to the surface. Large increases in the computational efficiencies of both wave packet methods were achieved by taking advantage of the potential coupling matrices associated with both methods becoming sparser with increasing molecule–surface distance. For the present model problem and employing this increased sparseness at longer range, the SAWP method is faster than the CCWP method by a factor of 2. The potential usefulness of the SAWP method for dissociative chemisorption problems is discussed.
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.
The close‐coupling wave packet (CCWP) method has been adapted for performing calculations on molecule‐surface scattering with arbitrary angles of incidence. The method used involves a slight modification of the fast Fourier transform (FFT) technique for evaluating the action of the translational kinetic energy operator on the wave function, employing the shifting theorem of Fourier analysis. We present and compare results of CCWP and close‐coupling (CC) calculations on the He+LiF and H2+LiF systems using simple model potentials. The results presented establish the validity of the proposed technique and may be useful as benchmarks.
We present a new and more efficient implementation of a hybrid approach to computing the solution of scattering problems affected by resonances. In the computationally expensive part of the calculation, wave packet propagation is used to obtain the time-dependent wave function Ψ(t) up to some time τ at which direct scattering is over. This part is made efficient by using a recently introduced modification for the absorbing boundary conditions evolution operator which allows the use of real operator algebra if the initial wave function is chosen real. In the second part of the calculation, filter diagonalization is used to efficiently obtain the energies, widths, and expansion coefficients of resonances needed to describe the long time behavior of the scattering wave function. This part is made efficient by using a recently introduced algorithm which avoids the storage of energy-dependent basis functions. We demonstrate the application of the method to a two-dimensional reactive scattering problem. © 1997 American Institute of Physics.
Theoretical investigations of molecule–surface scattering are performed using the multiconfiguration time-dependent Hartree method. Rotational and diffractive inelastic scattering of a rigid diatomic molecule from a corrugated static surface is investigated. The numerical simulations concern a five-dimensional N2/LiF(001) model system for collision energies ranging from 60 to 300 meV. A correction scheme of the energy distribution of the initial wave packet allows the simulation to be started close to the surface, in a region where the interaction potential is not negligible. The analysis of the propagated wave packet is performed using a combined flux operator/complex absorbing potential approach to extract partially summed transition probabilities and average energy transfers to selected internal modes. The scattering mechanism is investigated in detail. The surface corrugation is seen to damp the quantum interferences in the rotational excitation process and to induce rotational excitation to the low excited rotational states. The diffraction process and the impact of the initial rotational state of the diatom, in particular its initial orientation with respect to the surface, are discussed. © 2001 American Institute of Physics.
We apply the mixed quantum/classical method based on the Bohmian formulation of quantum mechanics [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case of rotational diffractive surface scattering of a diatomic molecule. The rotation as well as the normal translational degree of freedom are treated classically while the two parallel degrees of freedom that account for the diffraction are treated quantum mechanically. The effects of treating some degrees of freedom classically are discussed in detail by comparing our novel approximate method to quantum wave packet results obtained by the multiconfiguration time-dependent Hartree method. © 2002 American Institute of Physics.
We emphasize the merits and the superiority of the most complete nondirect product representation in non‐Cartesian coordinates. Beyond the proper choice of basis set we show how to further optimize the spectral range in multidimensional calculations. The combined use of a fully pseudospectral scheme and the finite basis representation (FBR) as the primary space ensures the smallest prefactor in the semilinear scaling law of the Hamiltonian evaluation with respect to the FBR size. In the context of scattering simulations we present a simplified asymptotic treatment which increases the efficiency of the FBR‐based pseudospectral approach. An optimal 6D pseudospectral scheme is proposed to treat the vibrational excitation and/or dissociation of a diatomic molecule scattering from a rigid, corrugated surface, and serves to illustrate our discussion. A 5D numerical demonstration is made for the rotationally inelastic scattering of N2 from a model LiF surface. © 1994 American Institute of Physics.
We have investigated the performance of a fully close‐coupled wave packet method and its symmetry‐adapted version for a model problem of H2 scattering from LiF(001). The computational cost of the fully close‐coupled methods scales linearly with the number of rotation‐diffraction states present in the basis set, provided that the sparseness of the potential coupling matrix is taken into account. For normal incidence, the symmetry adapted version is faster than the conventional close‐coupling wave packet method by almost an order of magnitude. An extension of the method to more realistic molecule‐surface problems is considered. © 1995 American Institute of Physics.
The recently proposed coupled channel density matrix (CCDM) method for nondissipative dynamics [L. Pesce and P. Saalfrank, Chem. Phys. 219, 43 (1997)], is extended to open quantum systems. This method, which is the density matrix analogue of the coupled channel wave packet (CCWP) method in Schrödinger wave mechanics, allows for the solution of nuclear Liouville–von Neumann equations in more than one dimension including unbound modes. A semiphenomenological, Markovian, and trace-conserving dissipative model within the dynamical semigroup approach is suggested, and efficient numerical schemes for its implementation are presented. Using a two-mode model, we apply the dissipative CCDM method to the problem of vibrationally excited gas-phase hydrogen molecules, relaxing during the scattering from a cold, metallic, and nondissociative surface. The significance of a relaxation mechanism based on electron-hole pair creation in a metallic substrate is addressed. The dependence of the survival probability of the vibrationally excited molecules on the dissipative model parameters, on their initial translational energy, and on isotopic substitution is examined and rationalized on the basis of a simple classical kinetic model. © 1998 American Institute of Physics.
We apply two hybrid methods for solving scattering problems affected by resonances, to a four-dimensional reactive surface scattering system. In each method the solution of the problem is divided into two parts: a wave packet propagation, and a resonance calculation; results of the resonance calculation are used to extrapolate the long-time behavior of the system. In the first hybrid method, the propagation is by the multistep Chebyshev method, with calculation of resonances performed by the Lanczos method. In the second, the propagation is done using an implementation of the absorbing boundary condition (ABC) evolution operator, and the resonance calculation by filter diagonalization (FDG). Each method produces accurate scattering results in much less computation time than standard long-time wave packet propagation. The Chebyshev–Lanczos approach proves most capable for the calculation of resonances, but is computationally expensive. The ABC–FDG method is much cheaper to implement, but could not be made to extract accurate data for certain broad, overlapping resonances. This was overcome by propagating longer (still much shorter than for long-time propagation) to allow the elusive resonances time to decay. © 1998 American Institute of Physics.
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
The time-dependent quantum mechanical approach has emerged as a powerful and a practical computational tool for studying a variety of gas-phase chemical problems in recent years. In this report, we discuss the various developments that have made this possible with special emphasis on methodology and application to reactive scattering, photo-excitation processes and gas-surface interaction.
A new tool for solving nuclear Liouville-von Neumann equations for unbound problems in multi-dimensions, termed the Coupled Channel Density Matrix (CCDM) method, is introduced. In this method, applicable to situations where a subset of the active modes is “free” and another one “bound”, a mixed representation (coordinate space representation for the free, state representation for the bound degrees of freedom) is used for all operators, leading to a set of coupled low-dimensional Liouville-von Neumann equations. Various numerical investigations are carried out to characterize the performance of the method in its application to the non-dissipative inelastic scattering of H2 molecules and isotopomers from a Cu model surface.
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical‐like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
Quantum‐mechanical approximation methods and calculations for rotational transitions in molecule‐surface collisions are reviewed. The methods are analyzed with regard to predictions of several observable effects: (1) Large ΔJ (rotational quantum number) transitions, and rotational rainbow scattering. (2) The relation between rotational and diffractive transitions. (3) Scaling properties of the rotationally‐inelastic scattering amplitudes. (4) Trapping and resonances induced by rotational‐translational energy transfer. The methods examined with regard to some of these effects include: coupled‐channel calculations; the Sudden approximation with regard to both the rotational and the diffractive transitions, and the hard corrugated wall model. It is concluded that whilst available methods provide a qualitative description of the effects mentioned, quantitative treatment of real systems remains an open problem. The main difficulties in formulating a satisfactory quantitative model are examined. Finally, the article presents new results on molecular reorientation processes (ΔM J transitions) in molecule‐surface collisions. It is shown, using the Sudden approximation, that molecular reorientation probabilities should reflect sensitively on surface structural corrugation.
In this paper we advance a new approximation scheme for the dynamics of an N particle system, which is based on the expansion of the Heisenberg equations of motion in powers of h in the coherent state representation. The time evolution of the physical observables is generated classically by a Hamiltonian Hs, which is derived from the original Hamiltonian by adding 2N2 virtual particles to the system. These virtual particles are related to the dispersion of the quantum mechanical wave function, and reflect the (time dependent) sensitivity of the instantaneous positions and momenta xi, pi to the variation of the initial conditions. The approximate equations of motion preserve the commutation relations between the coordinates and the momenta and the total (quantum) energy. The present approximation may be adequate for the study of the dynamics of coupled anharmonic oscillators in the quasiperiodic regime.
A quantum mechanical calculation of the collision of H2 with the (001) face of LiF has been carried out and compared with recent experimental data in which rotational transitions were resolved. A simple model potential accounts quantitatively for many of the observed effects. Refined potential parameters are extracted and the effects of some of these parameters on rotational transitions are discussed.
The state sensitive detection technique of laser-induced-fluorescence has been employed to study inelastic collisions of NO molecules with Ag(l 111). The results show that direct collisions are predominant and that the effects of surface corrugation are small. Evidence is found for a rotational rainbow and for the production of rotational polarization.
Exact quantum-mechanical calculations are present for He scattering from one-dimensional models of disordered, mixed Xe + Ar overlayers. A time-dependent wavepacket approach is used with a recent technique for solving the Schrödinger equation. Results are given for several overlayers of different Xe : Ar concentration ratios. The dependence of scattering intensities on the disordered structures is discussed. The results provide a reference for testing approximations for scattering from disordered surfaces.
A quasiclassical trajectory study of N2 scattering from a rigid corrugated crystal surface was carried out. Rainbow maxima were seen in the mj distribution. The maximum value of product j at which mj, rainbows occur is found to increase linearly with the corrugation and decrease linearly with the hardness of the interaction potential.
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.
The evolution of wave packets under the influence of a He´non–Heiles potential has been investigated by direct numerical solution of the time-dependent Schro¨dinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈&psgr;(0)‖&psgr;(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.
In this paper we discuss the use of the total angular momentum representation in the close coupling-wave packet (CCWP-J) method for solving the time dependent Schro¨dinger equation for inelastic, nonreactive gas phase atom–diatom collisions. This enables the wave packet propagation for the relative motion to be reduced from three dimensions to one. The approach utilizes a close coupling expansion of the wave packet into subpackets labeled by quantum numbers for total angular momentum J, z-component of angular momentum m0, rotor angular momentum j, and orbital angular momentum l. The number of coupled subpackets is less than the number for the plane wave boundary condition CCWP method when J
A new time-dependent quantum-mechanical method which combines a coupled-channel expansion with a wave packet formalism is described. This method is applied to the study of rotationally inelastic scattering of H2 from a flat, rigid surface. Comparison with the results from a time-independent coupled-channel calculation shows the new method to be highly accurate.
Rotationally mediated selective adsorption scattering resonances are used to make an experimental and theoretical study of the laterally averaged interaction potential between HD and a weakly corrugated system, Ag(111). The experimentally observed resonances determine the vibrational levels of the HD/Ag(111) physisorption potential as a function of bound rotational state. These vibrational levels show J-dependent shifts due to the orientational anisotropy of the potential. Exact quantum scattering calculations using a full laterally averaged potential of the form V sub o(z,0) = v sub o (z) (1 + beta P sub 2 (cos theta)) have been carried out to obtain rotationally inelastic transition probabilities. Experimental and theoretical resonance energies are compared for two forms of v sub o(z), a Morse and a variable exponent potential, as a function of Beta, and are found to be very close to the first order perturbed energies of a free rotor in bound states of v sub o(z). Both potential forms give equally good fits to the data, yielding an optimum value of the asymmetry parameter, Beta approx. -0.05. The determination of Beta is relatively insensitive to small changes in the v sub o(z) well depth.
Using the stationary phase limit of the quantum mechanical full sudden S matrix for diatom‐corrugated rigid surface scattering of Proctor, Kouri, and Gerber [J. Chem. Phys. 80, 3845 (1984)] we obtain expressions for the rotational actions j and mj of an inelastically scattered diatom. By integrating over all points on the lattice, we reduce these to functions of the polar angles θ and ϕ only. It is found that j is a strong function of θ, but almost independent of ϕ, whereas the reverse is true for mj. Both j(θ) and mj(ϕ) are continuous plots which exhibit extrema known to produce rainbow behavior in inelastic gas‐phase scattering theory. We propose that this implies the existence of rainbows in the mj distribution, and show the dependence of these rainbows on various potential parameters, including corrugation, potential repulsion parameter, and lattice constant. The results explain earlier trajectory studies.
Exact close-coupling calculations for H2 rigid rotor scattering from a LiF (001) surface are presented. Using a standard model potential, rotation-diffraction transition probabilities are reported for a range of collision energies up to 0.7 eV and for various surface corrugations. The calculations are aimed to serve as a reference for approximate approaches. The exact transition probabilities are compared with existing quasiclassical trajectory calculations and with results obtained by the matrix-diagonalization sudden method. In agreement with previous approximate treatments it is found that the coupling between rotations and diffractions is only weak so that both can essentially be treated independently.
The measurement of the alignment and orientation of nitrogen scattered from Ag(111) is reported. Pulsed supersonic beam of nitrogen is scattered off a clean Ag(111) crystal and the scattered nitrogen is detected by two-photon resonant four-photon ionization.(AIP)
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.
Quantum close-coupling calculations are presented which verify that the rotational polarization observed in recent experiments on NO-Ag(III) scattering is due primarily to mj conservation. However a dramatic dependence of rotational transitions on initial mj in some surface collisions is also indicated by the calculations and may be experimentally observable.
Diffractive and rotationally mediated selective adsorption scattering resonances are reported for n-H2 p-H2 n-D2 and o-D2 on Ag(111). Small resonance shifts and line-width differences are observed between n-H2 and p-H2 indicating a weak orientation dependence of the laterally averaged H2/Ag(111) potential. The p-H2 and o-D2 levels were used to determine the isotropic component of this potential, yielding a well depth of ~ 32 meV.
The effect of varying the corrugation of the molecule-crystal potential surface in N2-crystal collisions has been investigated. Increasing the corrugation parameter, β, increased the average rotational energy transfer. In addition, large values of β allow quite marked deviation from mj conservation.
The orientation of the angular momentum of Nâ scattered from clean Ag(111) is determined by resonance-enhanced multiphoton ionization. The orientation is the net helicity or handedness of the sense of rotation, i.e., clockwise vs counterclockwise. The orientation of the scattered Nâ is measured along a direction perpendicular to the scattering plane. The degree and sign of the orientation is found to depend strongly on the final rotational quantum number J and on the final scattering angle. The results require that there are forces acting in the plane of the surface during the scattering. The observed behavior can be reproduced qualitatively by a conventional hard-cube, hard-ellipsoid model to which a tangential friction has been introduced to account for the in-plane forces. This produces a splitting of the rotational rainbow peak which leads to changes of sign of the orientation as a function of rotational quantum number. Thus, orientation measurements provide a unique probe of in-plane gas--surface forces.
The collision of a diatomic molecule with a solid surface (neglecting phonon excitation and molecular fragmentation) is formulated as a set of coupled linear, second-order differential equations. A model potential for the collision of H2 with the (001) face of LiF is proposed. Scattering intensities are computed including 29 diffraction processes for the ν = 0, J = 0 internal state and 5 diffraction processes for the ν = 0, J = 2 internal state (34 channels in all). By comparing with a calculation omitting rotationally excited states (that is, the 29 channels for ν = 0, J = 0), it is observed that most of the 0 ↠ 2 rotational excitation occurs at the expense of the 0 ↠ 0 specular reflection. The 29 channel basis set includes all diffractions through third order and some fourth-order diffractions as well. Based on previous calculations of He+LiF scattering, it is concluded that this basis is sufficiently large to provide an accurate description of all rotationally elastic diffraction processes, but must be augmented to take proper account of rotation excitation. Calculations are also carried out for D2+LiF collisions, and several of the trends observed experimentally for these two isotopes are also observed here.
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.
Rotationally mediated selective adsorption of HD on Pt(111) is examined theoretically using R-matrix scattering techniques. With a laterally averaged surface-molecule Morse potential interaction and for an anisotropic potential term transformed from H2, excellent agreement is obtained between the resonances and the first-order perturbed bound vibration-free rigid rotor energies.
We present a detailed investigation of a new effect in molecule/surface scattering, rotationally mediated selective adsorption, which has been first observed by Cowin et al. (J. Chem. Phys. 75 (1981) 1033) in rotationally inelastic HD/Pt (111) scattering. It is due to resonance between an asymptotically closed rotational channel and a vibrational bound state of the molecule/surface system. Exact close-coupling and diffractionally sudden calculations are performed for HD scattered from a rigid, flat and weakly corrugated surface. Both variation of the collision energy and variation of the incident angle are considered. In the latter case we include the averaging over the collision energy of the initial beam and compare qualitatively our results with the experiment. Taking into account the resonances below they j = 0 → 1 threshold we are able to explain all of the experimentally observed resonances and to assign them with the appropriate rotational-vibrational quantum numbers. The largest deviation between experimental and theoretical resonance angles is ~ 1.6° at a collision energy of E = 109 meV. An interesting interference effect is found in the case of weak surface corrugation. Although the diffractionally sudden approximation generally gives only poor results, it also reproduces this effect and provides a simple explanation of it in terms of the Breit-Wigner representation of the S-matrix.
The scattering of a diatomic molecule from a solid surface is analyzed within an impulsive collision approximation. It is shown that for a particular simple molecule-surface potential function not only does such an assumption permit the separation of the rotationally inelastic contribution from the total scattered intensity, but also it allows one to estimate the importance of translation-rotation energy transfer on the basis of just a few parameters which appear in the chosen diatom-surface potential model. The implications and limitations of the use of such a model in the analysis of scattering data are discussed.
We compare two methods for the computation of the time evolution of a Gaussian wave packet located in the well of a Morse potential. We find that the use of second-order expansion of the potential around the center of the packet leads to inaccurate results for localized wavefunctions. The error appears because Heisenberg principle is not properly incorporated in the equation of motion for the center of the packet.
A theoretical approach to collision-induced energy transfer is presented that allows the direct calculation of the translational energy distribution of the separating fragments. The energy spectrum can be calculated with high or low resolution and the target states are not explicitly used. The formulation is directly derived from the wavepacket picture of time-dependent scattering and the approach is in principle exact. As a demonstrative example a semiclassical implementation, using gaussian wavepackets, is applied to collinear atom-diatom scattering.
The mechanism of rotational energy transfer in energetic (E ≲ eV) NO/Ag(111) collisions is investigated in quantal rigid-rotor/static-surface calculations. Reasonable agreement with recent experimental results is obtained for the case of a strongly asymmetric interaction potential which leads to two well-separated rotational rainbows at low and at high Erot, respectively.
The collision of H2 and D2 molecules with the (001) face of LiF is studied theoretically using the close coupling formalism. The method includes excited states of the diatomic molecule but neglects all couplings to phonons or the possibility of fragmentation of the incident molecule. Using a simple model potential a series of calculations for comparison with experimental data is carried out. Excellent agreement is obtained with the trends obeserved experimentally, and the quantitative agreement is also good.
Rotational state distributions for NO scattered from Ag(111) have been measured over the incident kinetic energy range of 0.1 to 1.7 eV with use of rotationally cold molecular beam and laser-induced-fluorescence detection. The measured distributions divide into two parts: a low rotational state portion (J≲20.5) which is described by a Boltzmann distribution and a high-J portion which shows a broad structure tentatively interpreted as resulting from a rotational rainbow.
An NMR method for investigating strong hydrogen bonding in solids is demonstrated for the adduct, KF·(CH2CO2H)2. A procedure for analysing the 19F NMR FID for KF·(CH2CO2H)2 is described and the results for the geometry of the OH…F… HO hydrogen-bonded system are compared with those of the X-ray structure. The value of the H…F bond length is found to be 0.148±0.003 nm which is significantly shorter than that determined by X-ray analysis.
We use analytic models, valid for neutron scattering, and numerical calculations to examine a method proposed by Drolshagen and Heller for computing atom diffraction from surfaces. Our main concern is the manner in which the use of narrow packets, causing a partial coherence of the incident beam, might affect the differential intensity. We find that the demands of representing beam coherence correctly conflict with the requirements for the validity of the propagation scheme. We suggest some improvements that should increase the reliability of the method without affecting greatly its remarkable computational and conceptual advantages.
The rotational polarization produced by scattering a rotationally cold beam of NO from Ag(111) has been measured by laser-induced fluorescence. A strong rotational polarization perpendicular to the surface normal is observed. The degree of polarization depends strongly on final rotational state, incident energy, incident angle, and surface temperature.
A model is proposed for vibrational deexcitation of diatomic molecules by collisions with a solid surface. The expressions obtained are analyzed to yield insight into the collision dynamics and used to predict the rotational and translational energy distributions, and other properties of interest. The method is developed in the approximation of a stationary surface, and is closely related to a recent model for vibrational relaxation in atom–molecule collisions. From considerations based on the scales of the relevant energy spacings and coupling strengths applied to the vibrational, rotational, and diffraction states involved, the scattering equations are greatly simplified by several approximations. For a simple but realistic class of potentials, analytical expressions are obtained for the deactivation probabilities pertaining to all final translational–rotational channels. Using the expressions of the model, a detailed study is made of: (i) The rotational–translational energy distribution produced by the vibrational energy release, and its dependence on system parameters; (ii) isotope and collision‐energy dependence of the deactivation probabilities; (iii) scaling properties of the transition probabilities with regard to ΔJ = J′−J, the change in rotational quantum number. The model is applied numerically to collisions of vibrationally excited H2, D2, T2, HD with a noncorrugated surface over a wide range of energies. The most striking feature of the model results is that a highly dominant fraction of the vibrational energy goes into molecular rotation, the main channel being an almost resonant V–R process in all cases.
We present theory and numerical results for a new method for obtaining eigenfunctions and eigenvalues of molecular vibrational wave functions. The method combines aspects of the semiclassical nature of vibrational motion and variational, ab initio techniques. Localized complex Gaussian wave functions, whose parameters are chosen according to classical phase space criteria are employed in standard numerical basis set diagonalization routines. The Gaussians are extremely convenient as regards construction of Hamiltonian matrix elements, computation of derived properties such as Franck–Condon factors, and interpretation of results in terms of classical motion. The basis set is not tied to any zeroth order Hamiltonian and is readily adaptable to arbitrary smooth potentials of any dimension.
Comparisons of the reflection and diffraction of thermal energy beams of 3He, 4He, H2, and D2 from the (001) surface of LiF are reported. Intense diffracted beams, including those of the second order, result from a freshly cleaved LiF surface without heating, and large exposures of H2O vapor do not reveal any measurable change in either the intensity, normalized to constant incident beam intensity, or location of such beams. The intensity and location of the individual beams are examined, taking into account the influence of adjacent beams. The locations of beams not so influenced display good agreement with the predictions of simple diffraction theory. The shapes and intensities of the diffracted beams are compared for the four gases with reference to possible inelastic effects. The decreasing intensities of diffracted beams for the same incident de Broglie wavelength are found to be in the order 4He, H2, and D2, i.e., in the order of increasing rotational energy coupling with the surface; 3He diffracted beams have intensities approximately the same as for 4He. The order of decreasing specular intensity, on the other hand, changes, depending on angle of incidence, but 3He is consistently more intense than for 4He, implying a mass collisional dependence. Preferentially scattered beams, similar to those reported for the heavier rare gases, are also evident although the presence of these beams only becomes apparent when conditions are such that the influence of other beams (diffracted and specular) is small. Selective adsorption studies reveal the presence of one or more bound energy states for each of the gases. The energy states for 3He and 4He are found to be significantly different, which is consistent with the results for the adsorption of these gases on charcoal. The energy states for H2 and D2 are identical, within the experimental error, and considerably greater in magnitude than for either 3He or 4He.
Performing the classical limit of the coordinate‐representation‐sudden approximation of Gerber et al. [J. Chem. Phys. 73, 4397 (1980)], we discuss rainbow effects in diatom‐surface scattering. Under special conditions, which are stated in this article, rainbows can be classified into surface rainbows and rotational rainbows. The latter are expected to be common features of diatom‐surface scattering provided: (i) the collision is impulsive and (ii) many rotational states are energetically open. Simple analytic expressions for the rainbow states are derived using a repulsive model potential and the dependence on collision and potential parameters is discussed. The predictions are all substantiated by calculations performed within the sudden approximation and using this model potential.
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.
We measured the rotationally inelastic diffractive scattering probabilities for an HD beam colliding with a smooth Pt(111) surface. These large T → R inelastic probabilities were measured as a function of incidence angle for a 110 meV beam energy. The results are in excellent agreement with a simple physical model of an eccentrically weighted sphere colliding with a hard wall, with an attractive well depth of 55±10 meV. The numerical ‘‘GR’’ method of Garcia and co‐workers was found superior to an eikonal method in solving this rotational quantum boundary value problem.
In this paper we describe the application of the close coupling wave packet (CCWP) method to the study of HD and H2 scattering from a Ag(111) surface. The presence of rotationally mediated selective adsorption (RMSA) resonances at low collision energies for the HD/Ag(111) system requires that the scattering wave function be propagated for extremely long times to allow the resonant states to decay. A new procedure is presented for obtaining both the magnitude and phase of the S‐matrix elements from the scattered wave packet which drastically reduces the size of the spatial grid required. Both the transition probabilities and collision lifetimes obtained from CCWP calculations are in excellent agreement with those predicted by the CC method, demonstrating that both the moduli and phases of S‐matrix elements can be obtained over a range of collision energies from the propagation of a single wave packet.
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.
Close‐coupled calculations of transition probabilities for rotationally inelastic scattering of NO(X 2Π) by a rigid, uncorrugated Ag(111) surface are presented. These calculations explicitly include the two adiabatic potential energy surfaces of V+ and V− symmetry, which are required in a fully quantum mechanical treatment of the collision dynamics. This enables us to provide the first theoretical description of the dependence of the transition probabilities on the spin–orbit and Λ‐doublet states of the scattered molecules, which are a direct manifestation of the open‐shell character of the 2Π electronic ground state of NO. A comparison is made with the results of experiments by Luntz, Kleyn, and Auerbach at IBM, and Zare and co‐workers at Stanford.
We investigate the suitability of the Wigner method as a tool for semiclassical dynamics. In spite of appearances, the dynamical time evolution of Wigner phase space densities is found not to reduce to classical dynamics in most circumstances, even as h→0. In certain applications involving highly ’coherent’ density matrices, this precludes direct h‐expansion treatment of quantum corrections. However, by selective resummation of terms in the Wigner–Moyal series for the quantum phase space propagation it is possible to arrive at a revised or renormalized classicallike dynamics which solves the difficulties of the direct approach. In this paper, we review the Wigner method, qualitatively introduce the difficulties encountered in certain semiclassical applications, and derive quantitative means of surmounting these difficulties. Possible practical applications are discussed.
A time‐dependent picture of vibrational Raman scattering in the weak field limit is presented. From this viewpoint we can separate the static effects, due to the coordinate dependence of the electronic transition dipole, from the dynamic effects that arise from wave packet propagation on the Born–Oppenheimer surfaces. Away from resonance, the energy uncertainty relation gives the propagation time necessary to obtain the cross section as being inversely proportional to the mismatch of the excitation frequency with the excited surface. The wave packet, given by the initial vibrational wave function times the transition dipole, hardly moves on the excited surface when the excitation frequency is far from resonance. As the excitation frequency is tuned closer to resonance, the propagated wave packet samples a larger portion of the surface. Using the short time approximation to the propagator, we obtain formulas for the cross section that are applicable for Raman scattering by polyatomics. The short time approximation is expected to be good away from resonance independent of the nature of the surface, and also on resonance with a repulsive surface. For an attractive surface, the approximation gives the average resonant cross section useful in the case when the vibrational structures cannot be observed.
A time‐dependent semiclassical method for generating energy‐dependent photodissociation partial cross sections is presented. The method is based on the Wigner equivalent formulation of quantum mechanics with the semiclassical limit arising from one dynamical approximation: the replacement of the quantum Liouville operator by its h→0 limit. The results of the present scheme for the collinear dissociation of ICN on a single dissociative surface are compared to those obtained from a distorted‐wave analysis and a semiclassical wave packet propagation. A model calculation employing standard trajectory techniques indicates that the present method has several distinct advantages over the traditional quasiclassical approach.
A time dependent wave packet approach to gas–surface scattering is presented. This semiclassical method is based on Gaussian wave functions whose average positions and momenta are those of classical trajectories. The initial states are represented as superpositions of Gaussian wave packets. These wave packets are propagated individually and after the collision, the scattering information is extracted by projecting them onto the final states in a coherent way, according to the initial expansion coefficients. The powerful new approach allows the treatment of surface imperfections as well as the inclusion of more degrees of freedom of the surface or the gas particle. The accuracy of the present semiclassical method is tested by a comparison with exact quantal calculations for He–LiF diffractive scattering and in most cases excellent agreement is found. In addition, the wave packet approach is used to calculate diffraction probabilities at high energies and to examine the interference structure of the scattered particles as a function of the size of the surface from which the scattering occurs.
We have probed the angular momentum orientation of N2 scattered from cold Ag(111) when the N2 approaches the surface along the surface normal. Using resonance enhanced multiphoton ionization (REMPI) and pulsed molecular beam techniques, we are able to probe the flux backscattered along the surface normal. In accordance with the restrictions on cylindrically symmetric systems, the molecules backscattered along the surface normal have no angular momentum orientation nor does the entire scattered flux integrated over all exit angles. However, for detection away from the surface normal, we observe substantial angular momentum orientation; the degree and direction of orientation depends upon the rotational state being probed. Molecular dynamics calculations reproduce the experimental results semiquantitatively. The calculations show that for N2 incident along the surface normal, the exit angle is largely determined by the two‐dimensional impact parameter of the molecule within the crystal unit mesh. However, the final rotational state, orientation, and alignment are determined largely by the molecular orientation geometry of the N2 during the collision. In essence, we have found a dynamical process which can partially differentiate between the two hidden initial conditions in a gas–surface collision: the two‐dimensaional impact parameter and the molecular orientation geometry.
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.