
Resonances
In Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics
Resonances, Copyright, ACS Symposium Series, FOREWORD
In Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics
M. Joan Comstock
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PREFACE
DONALD G. TRUHLAR
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GENERAL DISCUSSIONS
Roles Played by Metastable States in Chemistry
JACK SIMONS
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Metastable states are important in chemistry for reasons which relate to the fact that such states have finite lifetimes and finite Heisenberg energy widths. They are observed in spectroscopy as peaks or resonances superimposed on the continua in which they are buried. Their fleeting existence provides time for energy transfer to occur between constituent species which eventually become separated fragments. It is often the rate of such intrafragment energy transfer which determines the lifetimes of resonances. The theoretical exploration of metastable states presents special difficulties because they are not discrete bound states. However, much of the machinery which has proven so useful for stationary electronic and vibrational-rotational states of molecules has been extended to permit resonance energies and lifetimes to be evaluated. In this contribution, examples of electronic shape and Feshbach, rotational and vibrational predissociation, and unimolecular dissociation resonances will be examined. Finally, a novel situation will be treated in which the energy transfer dictating the decay rate of the metastable species involves vibration-to-electronic energy flow followed by electron ejection.
Direct Variational Methods for Complex Resonance Energies
C. WILLIAM McCURDY
In contrast to earlier complex coordinate methods which require a specific analytic continuation of the Hamiltonian, complex basis function methods for resonances rely on the existence of a complex variational principle for complex resonance energies, Er(R) - iΓ(R)/2, and are thus considerably more general, flexible and successful than their antecedents. A summary is presented of several methods which have appeared in the literature in this context with a view to displaying their similarities and common conceptual grounding. Those methods include: (1) complex SCF, (2) complex stabilization (CI), (3) the saddle-point coordinate rotation methods, and (4) analytic continuation of stabilization graphs. The last of these approaches requires only real-valued eigenvalue calculations, but nonetheless yields complex resonance energies directly.
Complex Energy Quantization for Molecular Systems
R. LEFEBVRE
The method of deriving resonance energies from propagations and a matching of the solutions of a set of coupled equations with appropriate boundary conditions is briefly presented. Complete or incomplete complex rotation of the interfragment coordinate gives the benefit of allowing the use of the same boundary conditions as for a bound state. The extension of propagation and matching to arbitrary paths in the complex plane is also described and an application is made to the treatment of electronic non adiabatic transitions.
Approximate Quantum Approaches to the Calculation of Resonances in Reactive and Nonreactive Scattering
JOEL M. BOWMAN - ,
KI TUNG LEE - ,
HUBERT ROMANOWSKI - , and
LAWRENCE B. HARDING
Reduced-dimensionality, quantum differential cross sections and partial wave cumulative reaction probabilities are presented for resonant and non-resonant scattering in the H+H2(v=0) reaction. The distorted wave Born approximation is tested against previous complex coordinate calculations of resonance energies and widths for a model van der Waals system. This approximation is subsequently used to obtain resonance energies and widths for the HCO radical using an approximate scattering path hamiltonian based on an ab initio potential energy surface. Several advantages of this hamiltonian for addition reactions are discussed.
ELECTRON-MOLECULE SCATTERING AND PHOTOIONIZATION
Resonances in Electron-Molecule Scattering and Photoionization
B. I. SCHNEIDER - and
L. A. COLLINS
The development of reliable theoretical models for calculating the decay of quasi-stationary states of molecular systems has become an important endeavor for theoretical chemists. The understanding and analysis of a wide variety of physical and chemical phenomena depend on a knowledge of the behavior of these states in both collisional and photoionization problems. In this article we describe the theory and calculation of these cross sections using our Linear Algebraic/Optical Potential method. The theory makes optimal use of the numeriical methods developed to solve large sets of coupled integral equations and the bound state techniques used by quantum chemists. Calculations are presented for a representative class of diatomic and triatomic molecules at varying levels of sophistication and for collisional and photoionization cross sections.
Dynamics of Molecular Photoionization Processes
D. L. LYNCH - ,
V. McKOY - , and
R. R. LUCCHESE
Increasing amounts of data have shown that shape resonances play an important role in molecular photoionization. These resonances can lead to significant deviations of the vibrational branching ratios from Franck-Condon predictions and to vibrational state dependence of the photoelectron asymmetry parameters. They are one-electron in nature and their properties are determined primarily by the molecular core potential. These features suggest that their role in photoionization should be studied using realistic molecular potentials and photoelectron continuum states. We will first discuss the relevant aspects of the method we have developed for studying the electronic Hartree-Fock continuum states needed in molecular photoionization. We will then present the results of applications of this approach to resonant photoionization in several molecules including N2, CO, CO2, C2H2, and C2N2. Our emphasis will be on a comparison of these results both with experimental data and other theoretical predictions.
Molecular Photoionization Resonances
A Theoretical Chemist's Perspective
P. W. LANGHOFF
Progress is reported in theoretical studies of molecular partial-channel photoionization and electron-impact ionization cross sections, with particular reference to clarification of the origin and nature of resonances that appear in certain final-state molecular symmetries. These features are attributed to the presence of virtual valence orbitals, largely of σ* character, above ionization thresholds where they are merged into photoionization continua. A quantitative (Stieltjes) theory is described for computational studies of cross sections in both discrete and continuous spectral regions, and illustrative examples are reported. Three-dimensional graphical representations of Stieltjes orbitals in molecular point-group symmetry aid in identification and interpretation of resonance contributions as the σ* orbitals of Mulliken.
Shape Resonances in Molecular Fields
J. L. DEHMER
A shape resonance is a quasibound state in which a particle is temporarily trapped by a potential barrier (i.e., the "shape" of the potential), through which it may eventually tunnel and escape. This simple mechanism plays a prominent role in a variety of excitation processes in molecules, ranging from vibrational excitation by slow electrons to ionization of deep core levels by X-rays. Moreover, their localized nature makes shape resonances a unifying link between otherwise dissimilar circumstances. One example is the close connection between shape resonances in electron-molecule scattering and in molecular photoionization. Another is the frequent persistence of free-molecule shape resonant behavior upon adsorption on a surface or condensation into a molecular solid. The main focus of this article is a discussion of the basic properties of shape resonances in molecular fields, illustrated by the more transparent examples studied over the last ten years. Other aspects to be discussed are vibrational effects of shape resonances, connections between shape resonances in different physical settings, and examples of shape resonant behavior in more complex cases, which form current challenges in this field.
Temporary Negative Ion States in Hydrocarbons and Their Derivatives
K. D. JORDAN - and
P. D. BURROW
Electron scattering experiments, in particular electron transmission spectroscopy (ETS), have provided a wealth of data on the temporary negative ion states of polyatomic molecules. Following brief introductions to the transmission technique and to the characteristics of resonances, we examine the spectra of several "simple" unsaturated hydrocarbons and discuss shape resonances arising from the temporary occupation of π* orbitals and, in the case of ethylene, the Feshbach resonances associated with Rydberg orbitals. The importance of long-range interactions in anion states is illustrated with data in several non-conjugated dienes. The classification of higher lying resonances in benzene is discussed with regard to their "shape" and "core-excited" characteristics. Finally we examine resonances associated with σ* orbitals and the decay of such states in the dissociative attachment channel.
Negative Ion States of Three- and Four-Membered Ring Hydrocarbons
Studied by Electron Transmission Spectroscopy
ALLISON E. HOWARD - and
STUART W. STALEY
Electron transmission spectra have been obtained for the C3-C6cis-cycloalkenes (3-6) and cycloalkanes (7-10). The somewhat low value for the first negative ion state of 3 (1.73 eV) can be understood on the basis of compensating effects due to the short C=C bond and the "reversed" polarity of 3 relative to 4-6. The lowest # resonances observed in 3, 4, 7, and 8 have been assigned on the basis of ab initio 6-31G molecular orbital calculations and by consideration of the components of the angular momentum associated with each resonance.
Anion Resonance States of Organometallic Molecules
JUDITH C. GIORDAN - ,
JOHN H. MOORE - , and
JOHN A. TOSSELL
The technique of electron transmission spectroscopy has only recently been used to measure the energies of anion states of organometallics, following a long and very significant history of application to the study of di- and triatomics and unsaturated hydrocarbons. Two classes of investigation have been carried out: the study of transition metal compounds such as Cr(CO)6, Fe(CO)5, ferrocene, and other metallocenes; and the study of organic species such as benzene on which organometallic and other main group organo ligands have been substituted. Theoretical calculations, the most successful of which employed the multiple scattering Xα formalism, have been used to describe the scattering process and assign anion states in the transition metal compounds as well as model compounds such as SiH4. Most recently mass spectrometric techniques have been used to identify fragments arising from dissociative anion states.
Vibrational-Librational Excitation and Shape Resonances
In Electron Scattering from Condensed N2, CO, O2, and NO
L. SANCHE - and
M. MICHAUD
Low-energy (0.2-30 eV) electron scattering from multilayer films of N2, CO, O2, and NO, condensed on a metal substrate near 20 K, has been investigated using hemispherical electron spectrometers and one-dimensional multiple scattering theory. Comparisons of experimental electron energy loss spectra with those generated theoretically indicate that the former are composed of peaks which result from single and multiple vibrational losses convoluted with multiple librational phonons having a mean energy of about 8 meV. Strong and broad peaks in the energy loss functions of the v=l to 3 vibrational levels of ground state N2, CO, and O2 are interpreted to arise from the formation of transient anions. In N2 and CO, all of the gas-phase shape resonances are observed, whereas in O2, only the 2Πu and 4Σ-4states seem to exist. The 2Πg state of N2- exhibits vibrational structure and a relaxation shift of 0.7 eV in the solid. Nitric oxide dimerizes upon condensation and we observe two broad resonances at 11.6 and 14.2 eV in the ν1 and ν5 decay modes of the dimer. They appear to be associated with the splitting of a single NO state via "through space" orbital interaction.
VAN DER WAALS COMPLEXES
Vibrational Predissociation of Small van der Waals Molecules
ROBERT J. LE ROY
The nature of vibrationally and rotationally predissociating states of atom-diatom Van der Waals molecules and the fundamental considerations governing their predissociation are discussed. Particular attention is focussed on the influence of the potential energy surface and the information about it which might be extracted from accurate measurements of predissociation lifetimes. Most of the results discussed pertain to the molecular hydrogen-inert gas systems, and details of previously unpublished three-dimensional potential energy surfaces for diatomic hydrogen with krypton and xenon are presented.
Complex-Coordinate Coupled-Channel Methods for Predissociating Resonances in van der Waals Molecules
SHIH-I CHU
Complex-Coordinate Coupled-Channel (CCCC) methods are presented for the accurate and efficient treatment of the resonance energies and widths (life-times) of multichannel rotationally predissociating van der Waals (vdW) molecule resonances. Algorithms for dealing with the complex scaling of numerical and piecewise analytical potentials are also presented. The CCCC methods for vdW complexes are formulated in both the space-fixed (SF) and the body-fixed (BF) coordinates. The SFCCCC method is more appropriate for the treatment of weak-coupling complexes (such as Ar-H2), whereas the BFCCCC method is better for strong-coupling complexes (such as Ar-HCl). These methods have been applied successfully to a number of vdW molecules, using reliable potential surfaces determined by experiments. In particular, the predicted widths for Ar-HD are in good agreement with the recent experimental data of McKellar.
Vibrationally Excited States of Polyatomic van der Waals Molecules
Lifetimes and Decay Mechanisms
W. RONALD GENTRY
The existing data on the infrared photodissociation of polyatomic van der Waals molecules are reviewed from a dynamical perspective. Correlations are examined between the lifetimes derived from the homogeneous linewidths for photodissociation and the structural and dynamical parameters of the various molecular systems. Two conclusions emerge: that the linewidths are determined principally by microscopic dynamics rather than statistics, and that the dynamics are dominated by vibrational-rotational coupling instead of vibrational-translational or vibrational-vibrational coupling. On this basis, the hypothesis is offered that the linewidth-derived lifetimes correspond not to the vibrational predissociation rates, but to the rates of vibrational relaxation within the metastable complexes by coupling of the initial vibrational state to those van der Waals modes which might best be described as internal rotations and/or librations.
Photodissociation of van der Waals Molecules
Do Angular Momentum Constraints Determine Decay Rates?
M. P. CASASSA - ,
COLIN M. WESTERN - , and
KENNETH C. JANDA
Experimental data pertinent to the vibrational predissociation mechanism of two types of van der Waals complex are presented and discussed. First, variations in the infrared band shape for excitation of the ethylene out-of-plane wag, ν7, in the series of molecules (C2H4)2, C2H4:HF, C2H4:Ne are discussed in terms of structure and relaxation mechanisms. Second, rotationally resolved laser excited fluorescence spectra for NeBr2 and NeCl2 are presented. There is a strong dependence of decay rate on molecular structure. Relaxation lifetimes vary from less than 10-12 s for C2H4 dimer to greater than 10-5 s for NeCl2. Trends observed are qualitatively predicted by consideration of linear and angular momentum gap arguments. Other possible interpretations are discussed.
UNIMOLECULAR DYNAMICS
Classical, Semiclassical, and Quantum Dynamics of Long-Lived Highly Excited Vibrational States of Triatoms
R. M. HEDGES JR.,- ,
R. T. SKODJE - ,
F. BORONDO - , and
W. P. REINHARDT
Triatoms with doubly vibrationally excited predissociating states of exceptionally long lifetime are theoretically investigated using several techniques. For a two-degrees-of-freedom model of H2O fully converged quantum estimates of resonance lifetimes are made, confirming the correspondence principle expectation that exceptionally long lived states exist and display non-RRKM behavior in the sense that lifetimes do not always decrease with increasing energy above the dissociation limit. Using the quantum results as a benchmark, the validity of a Golden-Rule-type formula is demonstrated, and the formula is then applied for physically realistic values of frequency, anharmonicity, and well depth: States with lifetimes of up to 0.1 sec are found. The paper ends with presentation of preliminary adiabatic semiclassical estimates of resonance energies for HOD in two and three degrees-of-freedom models.
The Intramolecular Dynamics of Highly Excited Carbonyl Sulfide (OCS)
MICHAEL J. DAVIS - and
ALBERT F. WAGNER
The intramolecular dynamics of highly excited OCS has been studied within the framework of classical mechanics. One study involves the energy relaxation of OCS restricted to planar geometry. A second study investigates the unimolecular dissociation of OCS. It has been found that the relaxation of OCS is slow and appears to occur on two time scales. In addition, the relaxation still persists at 45 ps. Reasons for such an effect, as well as diagnostic techniques, are discussed. Since the causes for such effects persist above dissociation they lead to nonstatistical uni-molecular dissociation, which also shows a high degree of mode specificity.
BIMOLECULAR REACTIVE SYSTEMS
Vibrationally Bonded Molecules
The Road from Resonances to a New Type of Chemical Bond
JOACHIM RÖMELT - and
ELI POLLAK
Bimolecular collinear reactions roughly may be divided into two categories: those occurring on attractive potential surfaces and those occurring on repulsive ones. For the former, the collision process is thought to proceed via a complex mechanism, the long living complex being a result of an attractive well between reactants and products. For a collision occurring on a 'repulsive' potential energy surface the process is assumed to be 'direct' since there are no attractive forces that can hold the colliding partners together. In this case the potential energy surface typically has a saddle point between reactants and products.
Surprisingly, early numerical computations for quantal scattering on saddle point surfaces exhibited spikes in the reaction probabilities as a function of the energy (1-4). Levine and Wu (5) showed that these spikes may be interpreted as resonances as they are associated with long (in comparison to vibrational periods) time delays. Subsequent numerical studies
Bimolecular Reactive Collisions
Adiabatic and Nonadiabatic Methods for Energies, Lifetimes, and Branching Probabilities
BRUCE C. GARRETT - ,
DAVID W. SCHWENKE - ,
REX T. SKODJE - ,
DEVARAJAN THIRUMALAI - ,
TODD C. THOMPSON - , and
DONALD G. TRUHLAR
Several approximate methods for calculating resonance energies and widths for atom-diatom reactive collisions are discussed. In particular, we present resonance energy calculations by semiclassical and quantal vibrationally adiabatic models based on minimum-energy and small-curvature paths, by the semiclassical SCF method, by quantal SCF and configuration-mixing methods, and by close coupling calculations. We also present total width calculations based on analytic continuation by polynomials and Padé approximants of configuration-mixing stabilization graphs, and we present total width and partial-width calculations based on close coupling calculations and on the Feshbach formalism in reaction-path coordinates with a small-curvature tunneling approximation for adiabatic decay and a reaction-path-curvature coupling operator for nonadiabatic decay. The model calculations are judged by their agreement with the accurate close coupling calculations, and we also compare the resonance energies and total widths to values obtained semiclassically from resonant periodic orbits. To illustrate the methods we consider the collinear reactions H + FH → HF + H and D + FD → DF + D on the low-barrier model potential of Muckerman, Schatz, and Kuppermann and the collinear and three-dimensional H + H2 reactions on Porter-Karplus surface number 2. Finally we use an accurate potential energy surface for the three-dimensional H + H2 reaction to predict the energies of several series of observable resonances for a real system.
Atom-Diatom Resonances Within a Many-Body Approach to Reactive Scattering
DAVID A. MICHA - and
ZEKI C. KURUOGLU
The resonance scattering of atoms by diatomics can be described within a many-body approach that provides a link between the electronic structure and the quantum dynamics of chemical bonding. Introducing a basis of spin-adapted valence-bond functions, the system hamiltonian matrix is decomposed into sums of atom-atom pairs and intrinsic three-atom contributions. Rearrangement scattering cross sections are obtained from coupled equations for three-atom transition operators, which provide a definition of quasibound states corresponding to resonances. Coupled integral equations in two vector variables are analyzed into angular momentum components and solved to obtain K-matrix elements converged within an s-wave model. This provides positions and widths of resonances for H + H2, over a wider range of energies than previously investigated.
Resonances in the Collisional Excitation of Carbon Monoxide by Fast Hydrogen Atoms
LYNN C. GEIGER - ,
GEORGE C. SCHATZ - , and
BRUCE C. GARRETT
This paper presents a detailed study of resonances in the collisional excitation of CO by H atoms at 1-3 eV translational energy. These resonances are all associated with formation of the metastable species COH during collision. The dynamics is studied using both classical and quantum versions of the infinite order sudden (IOS) approximation, and the resonance energies are characterized using stabilization calculations. We find that the vibrationally inelastic transition probabilities show complex resonance structure at energies above 1.2 eV, with the lowest six resonance states of COH narrow enough to be characterized by stabilization methods. Each resonance energy varies with both the orbital angular momentum ℓ and the atom-diatom orientation angle γ. A complete mapping of the (ℓ,γ) dependence of these six lowest resonance states is determined, and the resonances seen in the scattering calculations are all assigned in terms of their CO and OH stretch quantum numbers. These assignments are confirmed by examining plots of the scattering wavefunctions. The resonance widths are studied and are found to vary with vibrational state, with ℓ, and with γ. The smallest calculated width is about 4 x 10-4 eV corresponding to a lifetime of 2 ps. Integral cross sections for vibrational excitation (rotationally summed) are found to be not very sensitive to the presence of resonances, with good agreement between quantum and classical IOS results and between IOS and quasiclassical trajectory results except close to threshold for each vibrational state. A qualitative study of the γ dependent probabilities suggests that the resonances should measureably influence rotationally inelastic cross sections for high rotational states. They should be most important in studies of the rotationally resolved differential cross sections.
Resonant Quasi-periodic and Periodic Orbits
For the Three-Dimensional Reaction of Fluorine Atoms with Hydrogen Molecules
C. C. MARSTON - and
ROBERT E. WYATT
Numerical methods are described for locating resonant quasiperiodic and periodic orbits in the 3D F+H2 reaction with J=0. A number of plots of both types of resonant orbit are presented. This is the first time that resonant orbits have been found for a non-collinear reaction. These orbits are then used in the arbitrary trajectory semiclassical quantization scheme of DeLeon and Heller. The lowest resonance energy predicted using this procedure is in good agreement with all available quantal and adiabatic semiclassical results.
Resonance Phenomena in Quantal Reactive Infinite-Order Sudden Calculations
Z. H. ZHANG - ,
N. ABUSALBI - ,
M. BAER - ,
D. J. KOURI - , and
J. JELLINEK
In this paper we discuss the resonance tuning hypothesis as an important mechanism whereby resonances are spread in the F+H2 and F+D2 reaction systems and examine whether the shift from backward to sideways scattering of the HF and DF products is a resonance signature. All results are obtained using the Muckerman 5 potential surface.
Dynamic Resonances in the Reaction of Fluorine Atoms with Hydrogen Molecules
D. M. NEUMARK - ,
A. M. WODTKE - ,
G. N. ROBINSON - ,
C. C. HAYDEN - , and
Y. T. LEE
The reactions of F + H2, HD and D2 were studied in high resolution crossed molecular beams experiments. Center-of-mass translational energy and angular distributions were determined for each product vibrational state. In the F + H2 reaction, the v=3 product showed intense forward scattering while the v=2 product was backward-peaked. These results, in contrast to the backward scattering of all DF product vibrational states from F + D2 at the same collision energy, suggest that dynamical resonances play an important role in the reaction dynamics of this system. In the F + HD reaction, the strong forward scattering of HF products and backward scattering of DF products is in agreement with the prediction of a stronger resonance effect for HF formation. The effect of the H2 rotational excitation and the reactivity of F(2P1/2) are also discussed.
Reactive Resonances and Angular Distributions in the Rotating Linear Model
EDWARD F. HAYES - and
ROBERT B. WALKER
We use the Bending-Corrected Rotating Linear Model (BCRLM) to investigate in detail the way in which resonances may affect the angular distribution of reaction products. Using a lifetime matrix method, we separate the resonant and direct parts of the S matrix, and from the direct part we obtain angular distributions in the absence of the resonance. When applied to the F+H2(v=0) → HF(v'=2)+H reaction on the M5 surface, at an energy for which the angular distribution is strongly sideways peaked, we find the resonance enhances the intensity of the sideways scattering, and shifts the sideways peak toward the forward direction. We also describe conditions for which angular distributions have a significant forward scattering component at the reaction threshold. For the F+H2(v=0) → HF(v'=2)+H reaction on an improved potential surface, there is forward scattering at threshold because a resonance builds in at large partial waves. For the He+H2+(v=3) → HeH+(v'=0)+H reaction, there are many overlapping resonances in all partial waves, and the reaction proceeds without a barrier. Reactivity from many partial waves at the reaction threshold results in significant forward scattering.
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