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Effects of Isolated Impurities on Atom Scattering from Crystalline Surfaces. Exact Quantum-Mechanical Calculations
Abstract
A recent method for time-dependent wavepacket scattering calculations is applied to He scattering from a Cu surface with isolated Ar impurities. Several effects are found: (i) broad tails superimposed on each diffraction spike; (ii) shallow impurity rainbow maxima; (iii) impurity—surface interference peaks. A sudden approximation is applied to the interpretation of the newly found features.
... Although it is a rather crude approximation, the hard-wall model is quite insightful because it allows for explaining and understanding on simple terms the reflection symmetry interference phenomenon [23,24,27,47,50,51] as well as the conditions leading to trapping. As in geometric optics, the key element is the interpretation of wave phenomena in terms of the phase difference arising from two different but equivalent paths, where by "equivalent" we mean that both leave the surface with the same deflection (outgoing) angle, even if their journeys close to the surface are quite different (actually, it is this difference that generates the phase difference). ...
Bohmian mechanics, widely known within the field of the quantum foundations, has been a quite useful resource for computational and interpretive purposes in a wide variety of practical problems. Here, it is used to establish a comparative analysis at different levels of approximation in the problem of the diffraction of helium atoms from a substrate consisting of a defect with axial symmetry on top of a flat surface. The motivation behind this work is to determine which aspects of one level survive in the next level of refinement and, therefore, to get a better idea of what we usually denote as quantum-classical correspondence. To this end, first a quantum treatment of the problem is performed with both an approximated hard-wall model and then with a realistic interaction potential model. The interpretation and explanation of the features displayed by the corresponding diffraction intensity patterns is then revisited with a series of trajectory-based approaches: Fermatian trajectories (optical rays), Newtonian trajectories and Bohmian trajectories. As it is seen, while Fermatian and Newtonian trajectories show some similarities, Bohmian trajectories behave quite differently due to their implicit non-classicality.
This is the story of a career in theoretical chemistry during a time of dramatic changes in the field due to phenomenal growth in the availability of computational power. It is likewise the story of the highly gifted graduate students and postdoctoral fellows that I was fortunate to mentor throughout my career. It includes reminiscences of the great mentors that I had and of the exciting collaborations with both experimentalists and theorists on which I built much of my research.
This is an account of the developments of exciting scientific disciplines in which I was involved: vibrational spectroscopy, molecular reaction mechanisms and dynamics, e.g., in atmospheric chemistry, and the prediction of new, exotic molecules, in particular noble gas molecules.
From my very first project to my current work, my career in science has brought me the excitement and fascination of research. What a wonderful pursuit!
Expected final online publication date for the Annual Review of Physical Chemistry, Volume 72 is April 20, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
The angular intensity distribution of He beams scattered from compact clusters and from diffusion limited aggregates, epitaxially grown on metal surfaces, is investigated theoretically. The purpose is twofold: to distinguish compact cluster structures from diffusion limited aggregates, and to find observable {\em signatures} that can characterize the compact clusters at the atomic level of detail. To simplify the collision dynamics, the study is carried out in the framework of the sudden approximation, which assumes that momentum changes perpendicular to the surface are large compared with momentum transfer due to surface corrugation. The diffusion limited aggregates on which the scattering calculations were done, were generated by kinetic Monte Carlo simulations. It is demonstrated, by focusing on the example of compact Pt Heptamers, that signatures of structure of compact clusters may indeed be extracted from the scattering distribution. These signatures enable both an experimental distinction between diffusion limited aggregates and compact clusters, and a determination of the cluster structure. The characteristics comprising the signatures are, to varying degrees, the Rainbow, Fraunhofer, specular and constructive interference peaks, all seen in the intensity distribution. It is also shown, how the distribution of adsorbate heights above the metal surface can be obtained by an analysis of the specular peak attenuation. The results contribute to establishing He scattering as a powerful tool in the investigation of surface disorder and epitaxial growth on surfaces, alongside with STM. Comment: 41 pages, 16 postscript figures. For more details see http://www.fh.huji.ac.il/~dani
Atomic beam techniques are presently being used in many branches of surface physics such as studies of the particle-surface physisorption potential, surface structure, surface phonons, nucleation and growth on metal and insulator surfaces, surface diffusion and accommodation and sticking of molecules. This review concentrates on diffractive phenomena from surfaces, which up to now were investigated mainly with helium. The theoretical background for diffraction calculations is outlined and representative examples of different applications are given. The main subjects covered are: structural determinations of chemisorbed and physisorbed systems, investigations of disordered surfaces, selective adsorption resonances, diffusion and nucleation studies and investigations of growth and phase transitions on surfaces. Diffraction results obtained with Ne, Ar, and are also summarized.
Total cross sections for He scattering from isolated imperfections on surfaces are calculated using the Sudden approximation, and in some cases also by a numerically exact, time‐dependent quantum‐mechanical wave packet method. Systems studied include: CO adsorbates on Pt(111); mono‐, di‐, and trivacancies on Pt(111). The main results are: (1) the incidence angle and energy dependence of the cross section for He/[Pt(111)+CO] are very sensitive to the CO distance from the Pt plane. Interactions with the adsorbate image have little effect on the cross section. (2) The cross sections for clusters of vacancies are given within 10% or better, by the geometric sum of the monovacancy cross sections, the latter being treated as circles centered at each monovacancy. (3) The dependence of the cross section on the energy is sufficiently sensitive to distinguish between the ‘‘electron density hole’’ and ‘‘electron density hump’’ models for vacancies and vacancy clusters. (4) The Sudden approximation compares well with the exact quantum‐mechanical results at typical experimental energies, when the incidence angle is not too far from the normal. These results indicate that experimental measurements of He scattering cross sections as a function of energy and incidence angle, combined with Sudden or wave packet scattering calculations, can provide detailed information on surface defects and their interactions with gas‐phase atoms.
He scattering from substitutionally disordered Xe+Kr monolayers is studied theoretically. The angular distribution of the scattered He, and lifetimes of scattering resonances at the surface, are calculated for different Xe:kr mixing ratios. In addition to Bragg diffraction, new intensity maxima are found. These are interpreted as rainbow effects (disorder rainbows), and may be very useful in characterizing disordered surfaces. Also, trapping resonances are found, which are sharply sensitive to the percolation thresholds for the mixtures. The results show He scattering can be a powerful tool in studying 2D percolation behavior.
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
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.
We present a procedure that numerically evaluates the scattering wave function. The solution to the time-independent Schro¨dinger equation is calculated by a novel combination of: (a) the Mo&slash;ller operator of scattering theory, (b) time-dependent wave packets whose shape is unconstrained, and (c) efficient wave packet propagation on a dynamically-adapted grid. The superposition of packets appropriate to the scattering boundary conditions yields the full wave function, from which scattering amplitudes are then obtained. Since the procedure does not make use of basis-set expansions, its computational cost is independent of the number of open channels. It explicitly calculates the wave function not only in the asymptotic region but also within the interaction region, so it allows one to evaluate additional information beyond the scattering amplitude, as well as the functional sensitivity of transition probabilities with respect to changes in the potential. Applications here are illustrated by two simple examples: one-dimensional tunneling through a potential barrier, and elastic scattering from a one-dimensional periodic surface (i.e., a two-dimensional scattering problem). Extensive applications to imperfect surfaces including sensitivity analysis are separately presented in another article.
Rotationally inelastic scattering of diatomic molecules from isolated adsorbates on a flat surface is studied, focusing on properties of ΔmJ transitions (defined with respect to a quantization axis normal to the surface). A simple model is used, treating the molecule as a rigid pair of hard spheres and the adsorbate as a hard hemisphere. The sudden approximation is employed for the collision dynamics, resulting in simple expressions for the transition probabilities. Calculations are reported using model parameters that correspond to N2 scattered from Ar adsorbates on a flat surface. The results show pronounced maxima of the transition probabilities for large ΔmJ values, associated with a rainbow effect in the ΔmJ variable. Since low-corrugation crystalline surfaces gives rise to very weak, low-order ΔmJ transitions only, the results suggest that experimental studies of ΔmJ transitions could provide a sensitive probe of isolated adsorbates on surfaces and of molecule-adsorbate interactions.
A pseudospectral code for general polyatomic molecules has been developed using Gaussian basis functions. As an example, the water molecule is studied using a 6‐31G∗∗ basis set. Quantitative agreement with conventional calculations is obtained for the equilibrium geometry, total energy, first ionization potential, and vibrational force constants. Timing results for a vectorized version of the code (run on a Cray X‐MP) indicate that for large molecules, rate enhancements of Hartree–Fock self‐consistent field calculations of order 103 can be achieved.
The sudden approximation is applied to invert structural data on randomly corrugated surfaces from inert atom scattering intensities. Several expressions relating experimental observables to surface statistical features are derived. The results suggest that atom (and in particular He) scattering can be used profitably to study hitherto unexplored forms of complex surface disorder.
Angular distributions have been measured for helium atoms scattering from isolated CO molecules chemisorbed on a Cu(001) surface as a function of incident beam energy between 9.4 and 100 meV and surface coverage from 1.3% ML to 9.3% ML. Up to five oscillations are clearly observed in the angular distributions. The parallel wave vector transfers of the peaks vary only slightly with incident energy and are independent of coverage up to 20% of a c(2×2) layer. New hard wall scattering calculations show that all of the distinct peaks observed can be explained by interference structures involving both Fraunhofer diffraction and illuminated face scattering from CO molecules with an approximate hard wall radius of 2.4 Å with no evidence of the classical rainbows predicted in several recent theoretical studies.
We present a study of energy transfer (kinetic to vibrational) in collisions of atoms with diatomic molecules adsorbed on the surface of a metal substrate, for hyperthermal collision energies (0.1 to 1.0 eV). In order to make the many-body problem computationally tractable, atomic motions are restricted to one spatial dimension and the combined diatomic-metal target is modeled by a linear chain of coupled harmonic oscillators, so that vibrations of the target can be solved analytically for any arbitrary number of atoms. The collision is described in the semiclassical limit appropriate for hyperthermal velocities: translation of the projectile is obtained from a classical trajectory, while vibration of the target is treated quantum mechanically. The intensity of scattered atoms is obtained from the time-correlation function of the semiclassical transition operator. As a result, the intensity is evaluated analytically without need of internal-state expansions, and it includes the quantum-statistical average over the distribution of initial phonon states at nonzero temperature. The theory is applied to He projectiles scattered fromOCPtx, OCNix, N2Wx, and from the pure metals. The results are presented in the form of energy-loss spectral simulated for a typical experimental detector of finite resolution. The calculations are done with realistic values of force constants and He-target potentials. Hence the one-dimensional model contains the basic vibrational features of the gas–solid interface, namely: a very large number of low-frequency modes involving collective vibrations of the target atoms and a few high-frequency modes whose atomic displacements are localized near the surface (the latter roughly correspond to the vibration of the free diatomic and to stretching of the diatom-substrate bond). The simulated spectra show rich structure due to many-quantum excitations of collective and localized modes. We show how the structure is related to the eigenfrequencies and eigenvectors of the target, and we examine how the contributions of each mode vary with collision energy and target temperature. We find that excitation of localized modes follow a Poisson distribution and only the first few transitions are significantly excited. In contrast, collective modes undergo many-quantum transitions characterized by a quasicontinuous and Gaussian distribution of energy transfer. We derive simple expressions that explain the amounts of energy transferred to each mode in terms of the physical parameters (force constants, atomic masses, temperature, thickness of the target, and strength of the potential). The simulations indicate that vibrational transitions of the adsorbed molecule can be resolved from the backgrond of substrate phonons using present technology. They also suggest that lattice phonon structure could be observed using a thin substrate film weakly coupled to an underlying support. We also find that the spectrum is quite sensitive to small variations of the projectile-adsorbate potential, so that scattering experiments can yield information about how intermolecular forces are changed by chemisorption.
The scattering of He atom from a single CO molecule adsorbed on a smooth metal surface is modeled with an ab initio average potential of He-CO interaction super-imposed on a perfect reflecting flat surface. Integral as well as differential cross sections are obtained from exact quantum mechanical calculations. Rayleigh oscillations are shown to be present in the energy dependence of the integral cross sections. Among many Fraunhoffer and reflection-symmetry oscillations in the differential cross sections, a single peak is identified as the rainbow oscillation. This parameter free calculation can explain the observed integral and differential cross sections better than the previously used hard hemisphere model.
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.
Calculations are reported on He scattering from clusters of different numbers of CO molecules adsorbed in an ordered way on flat Pt(111) surfaces. The sudden approximation is used to obtain both the integral cross sections for scattering by adsorbates and the angular intensity distribution of the scattered atoms. The cross section values have been discussed on the basis of the Comsa and Poelsema overlap approach that, already successfully tested for clusters of vacancies, has been found to hold good even in the case of molecular adsorbates as surface defects. The angular intensity distribution curves show Fraunhofer interferences and rainbow maxima, as in the case of isolated adsorbates, and periodic diffraction peaks caused by the ordered structure of CO molecules on the Pt surface. Basically it has been demonstrated that from these peaks it is possible to obtain information on the geometric structure of the adlayer if the adsorbates form a two-dimensional crystal on the surface.
On the basis of the sudden approximation, a theoretical analysis has been performed on He scattering from two CO molecules adsorbed on Pt(111) at different mutual distances dCO (10 au≲dco≲ 50 au) and for the incident collision wavenumber kz from 1.0 to 4.0 bohr−1. The values obtained for the cross section Σ show that Comsa and Poelsema's hypothesis of cross-section geometric overlap (GO) breaks down. In fact, in a dCO range of nearly 22 au, and at all collision energies, the Σ values are higher than twice the cross section of the isolated adsorbed CO. Obviously this finding is not consistent with the GO hypothesis: the corresponding overlap between the two CO should be negative and therefore meaningless.
We employ the multiple-scattering expansion of the transition operator to evaluate the role of multiple collisions in elastic scattering of atoms from adsorbates on solid surfaces at superthermal energies (E ≳ 0.1 eV). The adsorbatesurface system is treated as a rigid-body model in the present work. Mixed-species periodic overlayers are considered as well as two-layer systems simultaneously involving periodic and non-periodic configurations. The calculated scattering patterns show rich structure due to the interference between waves scattered from different adsorbates. The enhanced sensitivity provided by wavevectors k ≳ 10 Å−1 could be exploited for surface characterization by scattering experiments. We find that the single-collision approximation qualitatively reproduces the trends of the intensity with respect to final angles. Double-collision terms are evaluated with the pole approximation and found to play an important role in large transfers of parallel momentum.
We combine Gaussian wave packets and the coupled channel method to develop a theory of H2 diffraction and rotational excitation by collision with surfaces. This improves our previous work on H2 diffraction since it eliminates the mean trajectory approximation; it also extends Heller’s work to problems in which the dynamics require the creation of new packets which must be coupled to each other as they are propagated through the interaction region. The approximations involved in the above Gaussian wave packet can be removed by extending a method proposed by Fleck, Morris, and Feit, which propagates the Gaussian wave function exactly and efficiently.
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 scattering of He atoms from a CO molecule adsorbed on a Pt surface is studied theoretically by methods that include: (1) Numerically exact solutions of the time‐dependent Schrödinger equation for the scattered wavepacket; (2) The sudden approximation; (3) Classical trajectories. The methods are used to obtain detailed insight into the collision dynamics, and to predict and understand interesting features in the angular intensity distribution of the scattered atoms. The analysis and interpretation of the exact quantum results is facilitated by calculations of the probability current density of the scattered particles. Some of the main results are: (i) The angular intensity distribution exhibits nonspecular maxima of two types: Several of the peaks are rainbow effects induced by the adsorbate, while others (at angles nearer to the specular) are Fraunhofer diffraction interferences. Both types of peaks contain useful, largely complementary, information on adsorbate geometry and on the He/adsorbate interaction. (ii) The angular intensity distribution is quantitatively sensitive to the adsorbate distance from the surface, suggesting possible determination of that distance from experimental data. (iii) The corrugation due to the adsorbate leads to scattering resonances associated with temporary trapping of the scattered atom at the defect site. This is a new effect of potential importance for experimental studies of atom/defect interactions. The results obtained here suggest that He scattering from isolated adsorbates exhibits distinct, substantial effects, measurement of which should yield very useful data on the adsorbates and on their interactions with gas‐phase atoms.
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.
The angular intensity distribution for atom scattering from surfaces with isolated defects is calculated. For a two‐dimensional model time‐dependent wave packet calculations are carried out for potentials with different steepness and variable defect size and height. The influence of the potential softness is studied by comparison with results from a corresponding hard wall model. Within the hard wall model the Eikonal approximation (EA) and an improved Eikonal approximation (IEA) which accounts for double collision effects are tested against numerically exact results. While, in general, the EA works well only in the vicinity of the specular peak the IEA is quite accurate over the entire angular range. Finally, wave packet calculations are carried out for a model of a flat surface with isolated steps, fitted to represent a stepped Pt(111) surface. The results are compared to recent measurements of the incoherent elastic diffraction for this system. Basic measured features like a rainbow maximum and large angle interference oscillations are well reproduced by the calculations.
The scattering of a monoenergetic beam of atoms by a hard hemisphere on a hard flat surface is formulated quantum mechanically. Differential and integral cross sections are calculated as a function of incident beam energy and incident angle and compared with the familiar results of scattering from a free hard sphere. Two types of interference processes are identified. In addition to the previously discussed off-specular reflection symmetry interferences, Rayleigh oscillations are found in the integral cross sections. They result from the interference of specular scattering from the hemisphere and from the flat surface. Their phases are found to be independent of the angle of incidence when plotted versus the normal component of the incident wave vector. © 1997 American Institute of Physics.
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.
It was recently proposed that the sudden approximation should be a powerful tool for the calculation of the angular intensity distribution in high‐energy atom scattering from disordered surfaces. In the present study the sudden approximation is applied to scattering from one‐ and two‐dimensional models of: (1) Isolated adsorbed impurities on crystalline surfaces (Ar on Cu); (2) Mixed overlayers on an underlying surface (Xe+Ar mixtures on a smooth surface). The results are tested against numerically exact quantum‐mechanical wave packet calculations. Except for very low collision energies, the sudden approximation gives results of excellent quantitative accuracy for both types of noncrystalline surfaces. At low energies, several features of the intensity distribution are not produced correctly by the sudden: These are found to be due mainly to double collision effects. The accuracy and validity range of the method are discussed in the light of the results obtained in the test calculations.
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.
We explore numerically the behavior of a method of describing the time dependent quantum mechanics of a curve crossing system. The two nuclear wave functions corresponding to the two electronic states are each described by a Gaussian wave packet. The packet describing the incident state mimics the initial wave function, and the other packet is created by the time dependent Schrödinger equation. They are both propagated by using a variational method. The packets interact and we do not assume that they have a small width. Exploratory calculations are made for curve crossing dynamics at low kinetic energy above the barrier of the lowest adiabatic state, for tunneling, for multiple crossings, and for a curve crossing system which is strongly coupled to a harmonic bath whose motion is described by a mean trajectory classical Langevin method.
An improved version of the pseudospectral method is used to accurately solve the Hartree–Fock equations for the neon atom and the diatomic molecules H2 and Li2. Because the method eliminates two‐electron integrals and is amenable to efficient vectorization on a supercomputer, order of magnitude reductions in computation time can be expected for large polyatomic molecules (for which a straightforward generalization of the procedure described here exists). The present work demonstrates that accuracy comparable to conventional methods can be achieved by the pseudospectral technique.
Functional sensitivity analysis is used to study the effect of potential structure upon the elastic scattering of He atoms from a one‐dimensional surface. The calculations are implemented by computing the total scattering wave functions from a wave packet calculation by a Møller wave operator method. The functional sensitivities of the various diffraction probabilities for several angles of incidence and surface corrugation are studied. The method is extended to examine the role of potential structure for a surface with adsorbed impurities. It was observed that the various diffraction processes draw from local regions of the potential in very different ways. At high angles of incidence for back scattering and particularly for strong surface corrugation, the large protruding portions of the surface cast a ‘‘shadow’’ of lower dynamical sensitivity. Results of this type should ultimately be insightful for the inversion of experimental data to obtain the interaction potential.
The angular dependence of the elastic scattering of helium atoms from a low coverage (ΘCO = 0.03) of randomly distributed isolated CO molecules adsorbed on a Pt(111) surface has been measured over a wide range of incident helium atom beam energies between Ei = 5.25 and 62.1 meV. The results are compared with a close-coupling theory for a realistic He–CO potential. From a best fit of the experimental results the distance of the molecule from the surface and a modified gas phase He–CO potential are obtained. © 2000 American Institute of Physics.
The pseudospectral method for Hartree–Fock calculations is applied to the glycine molecule, a test case with 100 basis functions. Several algorithmic improvements are reported, including a Newton–Raphson convergence scheme, Fock matrix updating, a multigrid technique, and optional recalculation of integrals. The pseudospectral method is shown to accurately reproduce the Roothaan–Hall relative and total energies for three conformations of glycine. Timing results show the pseudospectral code to be substantially faster than conventional Hartree–Fock codes.
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.
A projected continuum formulation for time‐independent quantum scattering from a continuum of channels is presented and applied to atomic scattering from disordered surfaces. Projecting out the channel continua gives rise to explicitly constructed optical potentials for the coherently diffracted scattering amplitudes, while the continuum diffuse scattering contribution is obtained in terms of these wave functions. Numerical application is made to atomic scattering from weakly corrugated disordered surfaces, for which a truncated set of coupled equations can be used within which the theory is shown to be unitary. The occurrence of a new phenomenon of disorder‐induced sticking on rigid surfaces is demonstrated and comparison with phonon‐induced sticking is made. These new disorder‐induced sticking processes are shown to have very different energy dependence than phonon‐induced sticking, with strong resonant sticking and oscillatory behavior seen at low incident energies.
Molecular beam scattering experiments have emerged in recent years as an important tool for investigating structural disorder on surfaces. This paper gives a brief overview of the developing theory of scattering from disordered surfaces, which is essential for the interpretation of such experiments, and presents new results on this topic. A brief discussion is given of the methods developed for calculating atom scattering disordered systems (e.g. time-dependent quantum wavepackets; distorted-wave approximation; sudden approximation) and of simulations of disordered surface structures (by Monte Carlo or molecular dynamics). The paper focuses, however, on features of the scattering intensities and their relation to disordered surface structures. The main topics include: (1) The concept of cross section for He scattering by an isolated defect on a surface, and its application to the study of interactions between defects. Comsa and his coworkers developed this into a most powerful tool for studying surface defects and defect interactions. We examine the assumptions involved, and the information contained in the measured cross section on defect shape and geometry, and on He/defect potentials. (2) Angular intensity distribution for He scattering from isolated defects; defect rainbow and Fraunhofer effects. (3) He scattering from periodic, substitutionally disordered surfaces. New results are presented on He scattering from mixed Xe + Kr monolayers on Pt(111), comparing theory with the experiments of Comsa et al. The results indicate that these monolayers are indeed periodic and (almost) perfectly substitutionally disordered for all Xe: Kr ratios. Rainbow intensity features due to the substitutional disorder are calculated and analyzed. (4) He scattering from amorphous mixed monolayers, with application to Xe + Ar mixtures on Pt(111). (5) Atom scattering from liquid surfaces, e.g. structure and dynamics of liquid Hg probed by Ar scattering.The paper concludes by emphasizing some of the outstanding open problems in the field.
Exact quantum-mechanical calculations are reported on atom scattering from a crystalline surface with isolated impurities. The calculations are for He scattering from a one-dimensional model of a Cu surface with adsorbed Ar atoms. The difficulties of carrying out calculations on scattering from extended but non-periodic structures are overcome by using a time-dependent wavepacket approach. A recently developed method for solving the time-dependent Schrödinger equation is employed. Scattering intensities are given for several energies and incidence angles. Detailed insight is obtained on impurity effects on surface scattering. The main features are: (1) Broad intensity tails are superimposed on each diffraction spike. The width of the tails decreases with increasing diffraction order; (2) Shallow rainbow peaks arise, due to impurity induced local corrugation; (3) Weak intensity maxima arise due to interference between surface and impurity scattering. The intensities are somewhat sensitive to the position of the impurity within the surface unit cell. Physical interpretation of the effects is provided from exact calculations, and from a simple sudden approximation for the scattering intensities. It is argued that He scattering can be used to determine impurity locations on surfaces.
The scattering of He atoms from O2 molecules adsorbed on a flat surface is studied, and comparison is made of models in which the O2 is respectively perpendicular and parallel to the surface. The sudden approximation is used to calculate both the angular intensity distribution of the scattered atoms at fixed incidence energy, and the integral cross section for scattering by the adsorbate as a function of collision energy. Significant differences are found in both the cross sections and the angular intensities between scattering by the parallel and by the perpendicular adsorbate. The possible use of He scattering to study adsorbate orientation is discussed in the light of the results.
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.
We describe a stable and efficient approach to generating a reliable polyatomic grid for self-consistent electronic structure calculations using the pseudospectral method. Systematic convergence of spectroscopic constants to the Roothaan Hartree-Fock values as the grid size is increased is demonstrated. The approach offers an estimated 1-2 order of magnitude reduction over alternative numerical self-consistent field algorithms in the number of grid points per atom required to produce accurate results.
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 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
Differential cross sections have been measured for the scattering of helium atoms from isolated CO molecules on a Pt(111) surface. The cross sections reveal an oscillatory structure as a function of scattering angle extending to large momentum transfer on both sides of the specular peak. Using a hard hemisphere model to approximate the interaction potential, the data can be well reproduced by a hard core of radius about 2.5 Å. This result is compared to the charge density profile of the adsorbed molecule and also to the gas phase interaction potential, and in both cases good agreement with the experimentally predicted classical turning points is found.
The H2 + Cu(1 0 0) system was used to test the performance of a newly developed, symmetry adapted pseudo spectral (SAPS) wave packet method for performing six dimensional quantum dynamical calculations on reactive scattering of diatomic molecules from square surfaces. Four test calculations have been performed on two different DFT PESs to compare the performance of the new SAPS method with that of the symmetry adapted wave packet (SAWP) method and the ordinary pseudo spectral method (PS). The comparison has been made for the (v=0,j=4,mj=0) and the (v=0,j=0,mj=0) initial states of H2, for the normal incidence energy ranges of 0.3–0.9 and 0.3–1.15 eV, respectively. The results for (v=0,j=4,mj=0) H2 reacting on PES 4, a PES expanded in a limited number of symmetry adapted functions so that it is suitable for use in the SAWP method, show that the SAPS method is competitive with the SAWP method, even for this PES. The results for (v=0,j=0,mj=0) H2 reacting on PES 5, a DFT PES represented using the corrugation reduction scheme, show that for this initial state the SAPS method is superior to the PS method by a factor of 6. It is concluded that, in general, for a given state the SAPS method will be at least three times faster than the PS method, also using at least three times less memory.
The role of surface defects in elastic gas-solid collisions is investigated by means of a recently developed numerical procedure based on the Møller operator and wavepackets. Since the procedure explicitly evaluates the scattering wavefunction, it yields the probabilities for scattering from the given initial state into all possible final states as well as the sensitivity of the probabilities with respect to variations in the gas-surface potential. Probabilities and their functional sensitivities are calculated for a simulated Pt surface exhibiting various configurations of vacancies: isolated sites, “interacting” di-vacancies and closely-packed tri-vacancies. The functional sensitivities indicate which regions of the gas-defect-solid potential are most relevant to the scattering dynamics. The resultant physical insight should be ultimately helpful for inverting experimental data to obtain the interaction potential.
Results of our initial study of the use of parallel architecture super-computers in solving time-dependent quantum scattering equations are reported. The specific equations solved are obtained from the time-dependent Lippmann-Schwinger integral equation by means of a quadrature approximation to the time integral. This leads to a modified Cayley transform algorithm in which the primary computational step is a matrix-vector multiplication. Implementation has been carried out both for the MasPar MP-1 and the NCUBE 6400 parallel machines. The codes are written in a modular form that greatly facilitates porting from one machine architecture to another. Both parallel machines prove to be more powerful for this application than the serial architecture VAX 8650. Specific analysis of machine performance is given.
The dissociative adsorption of hydrogen on Ni(110) at low temperatures has been studied in great detail with He diffraction. With increasing hydrogen exposure, a c(2 × 6), a c(2 × 4), two different c(2 × 6) and a (2 × 1) phase corresponding to coverages of monolayers (ML) have been identified. The corrugation functions of all phases have been determined by intensity analyses of He-diffraction spectra using the hard-wall approximation. The analyses were performed in a model-free manner using symmetry-compatible Fourier representations of the corrugations, and the results were underpinned by additional calculations with the adatoms modeled as Gaussian hills. The best-fit corrugations of all phases up to 1 ML deliver direct pictures of the adatom configurations, and allow an absolute coverage determination on purely crystallographic grounds in good agreement with the relative coverages determined by flash-desorption measurements. The most important structural elements are long H-zig-zag chains along the close-packed Ni rows with the hydrogen atom sites near threefold coordination. The rather long-ranged lateral interaction between the hydrogen chains favors alternation of zig-zag with zig-zag elements in all submonolayer phases. The c(2 × 6) phase with constitutes an interesting intermediate as H-adatoms are obviously forced into twofold coordinated sites to form a distorted close-packed hexagonal pattern of H atoms. Thus, the phase jump of 2.5 Å of all zig-zag chains necessary to form the (2 × 1) where all close-packed Ni rows are covered with parallel zig-zag chains seems to become feasible via a compression of the c(2 × 6) with . Further hydrogen can subsequently be accommodated via an adsorbate-induced reconstruction of the Ni substrate. Improved diffraction experiments provide the full two-dimensional corrugation function for the final (1 × 2) structure which corresponds to a saturation coverage of ≅ 1.5 ML whereby probably hydrogen is adsorbed on the second Ni layer.
The patterns of angular intensity distribution in molecular scattering from ordered stepped surfaces are studied by the classical trajectory method. Using the stepped Cu(117) surface as a representative model, the following results are obtained: (i) There is a nonspecular intensity structure due to the steps corrugation. In the angular range where data exists, intensities and peak positions are in good accord with recent experiments. (ii) A structure that is a secondary rainbow pattern is found superimposed on the familiar surface rainbow. The effect is due to multiple scattering related to step geometry. It is predicted that the secondary rainbow effect will be very typical in high incidence‐angle scattering from ordered stepped surfaces. (iii) A new surface‐scattering resonance is predicted that is a consequence of the ordered steps structure. It is expected that these resonances (long‐lived trajectories) can be the basis of a significant surface‐trapping mechanism. Capture into such resonances (step‐gliding resonances) is possible even at incidence energies that are very high compared with the well depth of the atom–surface potential. Finally, a method is given by which the step height and step size may be readily estimated from measured angular intensity distributions. Application of this estimate to simulated data from calculations give satisfactory results for the step parameters.
The diffraction of He atoms from solid surfaces has been recently used to determine the electron distribution of electrons at surfaces. From this, the position of underlying atoms can usually be inferred. Recent developments which test the validity of the hard corrugated wall method to calculate diffraction intensities are discussed. Structural studies which have been carried out to date, on various surfaces with He beams, are tabulated.
A number of experimental results for the diffraction of low energy helium by closely packed and stepped copper surfaces is presented. Observation of selective adsorption resonances shows that the surface average of the interaction potential is the same for all crystal faces. The thermal attenuation of the scattered intensities is given by a modified Debye—Waller factor. These results are interpreted in terms of interaction potentials with a soft exponential repulsion and an attractive well using an exact method based on Neumann iteration of the transition matrix equation. Good agreement with experiment is obtained in several systems over a large range of scattering energies.
In this paper two different physical situations are considered which can be treated with the same method: a fluid adsorbate (disordered in the x, y plane) and a clean surface with random steps (disordered in the z direction). The hard corrugated wall model is used in the eikonal approximation; the differences between the two cases arise only from the different statistical properties of the two physical situations. The differential scattering probability is evaluated. For the fluid adsorbate the latter splits into a coherent (purely specular) contribution and an incoherent one (which is, in fact, weakly inelastic and related to classical diffusion on the surface). For stepped “rough” surfaces only incoherent scattering is present and the differential scattering probability for hexagonal lattices is given.
The scattering of an atomic beam from a randomly stepped surface has been calculated using the hard corrugated wall model. Under the basic assumption that scattering from the step edges may be neglected the scattering equation can be solved without any further approximation. The solution displays the usual diffraction peaks each of them is broaden by a term characterizing the step configuration. If the step distribution is ergodic and stationnary in space this term is the characteristic function of the difference of level between two point of the surface considered as a random variable. Statistical models for the step repartition at the surface are proposed and the scattering intensity is derived in a closed form. The main result is that the broadening of the peaks varies from no broadening to a maximum according as the interferences from waves reemitted by the various terraces are constructive or destructive. A comparison is made with previous experimental data from which an estimation of the average step separation can be drawn. The sensitivity of the atomic beam scattering to steps is found to be more than one step every one hundred crystal atoms.
By assuming a Lennard-Jones 12-6 potential between He and Li+, and He and F− in a LiF crystal, the specular and diffracted intensities of He from a LiF(001) crystal surface are obtained by direct integration of the coupled differential equations. Numerical results are obtained by using 3–29 partial wave models for glancing incidence at θi = 60–90°. The 11 partial wave model gives fairly good results in comparison with other models having more waves. Experiments of Williams in which the incident conditions are varied by rotating the crystal are discussed.
The effect of roughness of a crystal surface on the electron diffraction intensities in lowenergy or high-energy reflection diffraction experiments is calculated by use of a model in which steps of uniform height occur at random intervals in each of two perpendicular directions. It is shown that the intensity distribution for a perfectly flat surface is modified by a function having maxima at the positions of kinematical diffraction maxima with no refraction effect. A graph is presented which allows an estimate of this modification function to be derived for the conditions of surface roughness and incident beam coherence width appropriate for any experiment.
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.
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.
The angular intensity distribution of atomic beams scattered from partially disordered solid surfaces is investigated. Quantitative relations are established between correlation functions that describe the structural disorder, and several features of the scattered intensity distribution. To simplify the collision dynamics, the study is carried out in the framework of the sudden approximation, which assumes that momentum changes perpendicular to the surface are large compared with momentum transfer due to surface corrugation. Simple relations are obtained between the structural correlation functions and the scattering at near‐specular angles, at extreme off‐specular angles, and for directions close to that at which the surface scattering rainbow appears for the corresponding ordered lattice system. Detailed expressions for the case of step disorder are presented. The results suggest that molecular beam scattering should be a powerful probe of structural properties of disordered surfaces.
Two versions of the sudden approximation are introduced to decouple and solve the equations that describe atom- surface scattering with many open diffraction channels. Both approximations require a high incident beam wave number compared with the magnitude of the reciprocal space vector of the lattice. In this framework, simple explicit expressions are obtained for the observable diffraction intensifies, making calculations feasible even for systems with hundreds of open diffraction channels. Further considerable simplifications ensue when the approximations are specialized to the case of a Lennard-Jones-Devonshire potential, or to that of a weakly corrugated surface. The approximations were applied to the systems He/LiF(001); Ne/LiF(001) and Ne/W(110) and the results are compared with other calculations or with experiment. The sudden approximation is found to be of good accuracy in these cases.
- Tommei
- Leibsch