Advances in Chemical Physics

By K. Birgitta Whaley

The Advances in Chemical Physics series provides the chemical physics field with a forum for critical, authoritative evaluations of advances in every area of the discipline.

•    This is the only series of volumes available that presents the cutting edge of research in chemical physics
•    Includes 10 contributions from leading experts in this field of research
•    Contains a representative cross-section of research in chemical reaction dynamics and state of the art quantum description of intramolecular and intermolecular dynamics
•    Structured with an editorial framework that makes the book an excellent supplement to an advanced graduate class in physical chemistry, chemical physics, or molecular physics

• Language: English
• Publisher: Wiley
• Release date: Mar 26, 2018
• ISBN: 9781119375050

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Editorial Board

Kurt Binder, Condensed Matter Theory Group, Institut Für Physik, Johannes Gutenberg-Universität, Mainz, Germany

William T. Coffey, Department of Electronic and Electrical Engineering, Printing House, Trinity College, Dublin, Ireland

Karl F. Freed, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA

Daan Frenkel, Department of Chemistry, Trinity College, University of Cambridge, Cambridge, UK

Pierre Gaspard, Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium

Martin Gruebele, Departments of Physics and Chemistry, Center for Biophysics and Computational Biology, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Gerhard Hummer, Theoretical Biophysics Section, NIDDK-National Institutes of Health, Bethesda, MD, USA

Ronnie Kosloff, Department of Physical Chemistry, Institute of Chemistry, Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem, Israel

Ka Yee Lee, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA

Todd J. Martinez, Department of Chemistry, Photon Science, Stanford University, Stanford, CA, USA

Shaul Mukamel, Department of Chemistry, School of Physical Sciences, University of California, Irvine, CA, USA

Jose N. Onuchic, Department of Physics, Center for Theoretical Biological Physics, Rice University, Houston, TX, USA

Stephen Quake, Department of Bioengineering, Stanford University, Palo Alto, CA, USA

Mark Ratner, Department of Chemistry, Northwestern University, Evanston, IL, USA

David Reichman, Department of Chemistry, Columbia University, New York City, NY, USA

George Schatz, Department of Chemistry, Northwestern University, Evanston, IL, USA

Steven J. Sibener, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA

Andrei Tokmakoff, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA

Donald G. Truhlar, Department of Chemistry, University of Minnesota, Minneapolis, MN, USA

John C. Tully, Department of Chemistry, Yale University, New Haven, CT, USA

List of Contributors Volume 163

Millard H. Alexander, Department of Chemistry and Biochemistry, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20741-2021, USA

Elie Assémat, Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel;

Theoretical Physics, Saarland University, D-66123 Saarbrücken, Germany

Zlatko Bačić, Department of Chemistry, New York University, New York, NY 10003, USA;

NYU-ECNU Center for Computational Chemistry, New York University Shanghai, Shanghai 200062, China

Eric R. Bittner, Department of Chemistry, University of Houston, Houston, TX 77004, USA

Joel M. Bowman, Department of Chemistry and Cherry L. Emerson Center for Scientific Computations, Emory University, Atlanta, GA 30322, USA

Tucker Carrington Jr., Chemistry Department, Queen's University, Kingston, Ontario K7L 3N6, Canada

David C. Clary, Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, UK

Paul J. Dagdigian, Department of Chemistry, The Johns Hopkins University, Baltimore, MD 21218-2685, USA

Peter M. Felker, Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095-1569, USA

Samuel M. Greene, Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, UK

Hua Guo, Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, NM 87131, USA

Henrik R. Larsson, Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, Olshausenstraße 40, D-24098 Kiel, Germany;

Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel

Jianyi Ma, Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, Sichuan 610065, China

Shai Machnes, Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel;

Department of Theoretical Physics, Saarland University, D-66123 Saarbrücken, Germany

Apurba Nandi, Department of Chemistry and Cherry L. Emerson Center for Scientific Computations, Emory University, Atlanta, GA 30322, USA

Andrey Pereverzev, Department of Chemistry, University of Missouri-Columbia, Columbia, MO 65211, USA

Bill Poirier, Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Lubbock TX 79409-1061, USA

Chen Qu, Department of Chemistry and Cherry L. Emerson Center for Scientific Computations, Emory University, Atlanta, GA 30322, USA

Xiao Shan, Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, UK

Xiangjian Shen, Research Center of Heterogeneous Catalysis and Engineering Science, School of Chemical Engineering and Energy, Zhengzhou University, Zhengzhou 450001, People's Republic of China;

State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People's Republic of China

David Tannor, Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel

Minzhong Xu, Department of Chemistry, New York University, New York, NY 10003, USA

Xunmo Yang, Department of Chemistry, University of Houston, Houston, TX 77004, USA

Dong H. Zhang, State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People's Republic of China

Foreword

This 163rd volume of Advances in Chemical Physics is dedicated to John C. Light, late Professor of Chemistry at the James Franck Institute and Department of Chemistry at the University of Chicago, who passed away in Denver, Colorado, on January 18, 2016. This memorial volume provides 10 contributions by former students and colleagues that focus on some of the core areas in quantum dynamics of molecular systems that fueled many years of pioneering research by John.

Born in Mt. Vernon in 1934, John studied at Oberlin College, Harvard, and then Brussels, before joining the faculty of the University of Chicago in 1961. He remained at the University of Chicago until his retirement in 2006, where he also provided long-term service to The Journal of Chemical Physics as an Editor (1983–1997) and an Associate Editor (1997–2007).

John's research interests were broadly focused on quantum dynamics of chemical systems. Within this area, he addressed a diverse and constantly evolving set of chemical and physical problems, with an emphasis on developing groundbreaking analytical and numerical formulations that took advantage of the rapidly growing power of computers during his career. John's legacy includes his pioneering work in developing the modern quantum theory of reactive molecular collisions, which laid the foundation for the high-precision quantum scattering calculations of chemical reactions being made today. This work led to his introduction of the highly efficient discrete variable representation (DVR) for scattering problems (with Jim Lill and Greg Parker in 1982). Recognizing the potential of the sparsity and flexibility provided by this dual representation, John subsequently extended the DVR to the analysis of intramolecular dynamics where it revolutionized the study of multidimensional bound states of molecular systems, allowing for a numerically exact quantum treatment of highly excited states, floppy molecules, and molecular clusters that was previously inaccessible.

The articles contributed to this volume in memory of John Light address topics in quantum molecular scattering dynamics, phase-space theory, intramolecular dynamics, and electron transfer dynamics. These areas reflect the breadth and enthusiasm of John's interest in both chemical reaction dynamics and the broader science that this connects to. John was a great scientist, a leader in his field, and a wonderful and highly respected colleague for many in the Chemical Physics community. He was also an inspiring mentor and scientific role model for generations of students and postdocs. John's vision and gracious persona will be missed by us all.

K. Birgitta Whaley

Department of Chemistry

The University of California, Berkeley

Preface to the Series

Advances in science often involve initial development of individual specialized fields of study within traditional disciplines followed by broadening and overlap, or even merging, of those specialized fields, leading to a blurring of the lines between traditional disciplines. The pace of that blurring has accelerated in the past few decades, and much of the important and exciting research carried out today seeks to synthesize elements from different fields of knowledge. Examples of such research areas include biophysics and studies of nanostructured materials. As the study of the forces that govern the structure and dynamics of molecular systems, chemical physics encompasses these and many other emerging research directions. Unfortunately, the flood of scientific literature has been accompanied by losses in the shared vocabulary and approaches of the traditional disciplines, and there is much pressure from scientific journals to be ever more concise in the descriptions of studies, to the point that much valuable experience, if recorded at all, is hidden in supplements and dissipated with time. These trends in science and publishing make this series, Advances in Chemical Physics, a much needed resource.

The Advances in Chemical Physics is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics: a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.

Stuart A. Rice

Aaron R. Dinner

July 2017, Chicago, IL, USA

Chapter 1

Applications of Quantum Statistical Methods to the Treatment of Collisions

Paul J. Dagdigian¹ and Millard H. Alexander²

¹Department of Chemistry, The Johns Hopkins University, Baltimore, MD, 21218-2685, USA

²Department of Chemistry and Biochemistry, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20741-2021, USA

Contents

I. Introduction

II. Quantum Statistical Theory

III. Fine-Structure Branching in Reactive O(¹D) + H2 Dynamics

IV. Inelastic OH + H Collisions

A. OH + H Vibrational Relaxation

B. OH + D Isotope Exchange

C. OH + H Rotationally Inelastic Collisions

V. OH + O Reaction and Vibrational Relaxation

VI. Inelastic Collisions of the CH Radical

A. CH + H2

B. CH + H

VII. H + O2 Transport Properties

VIII. Conclusion

Acknowledgments

References

I. Introduction

In chemical kinetics, statistical theories were first developed to understand unimolecular reactions and predict their rates (see, for example, [1, 2]). In the field of molecular reaction dynamics, we would expect statistical models to be well suited to reactions proceeding through formation and decay of a strongly bound collision Examples would be the reaction of electronically excited atoms with c01-math-001 [C( c01-math-002 D), N( c01-math-003 D), O( c01-math-004 D), and S( c01-math-005 D), for instance]. Here, atom M inserts into the H−H bond with the subsequent formation of a transient HMH complex, which then decays to form MH + H products.

Statistical models for reactions involving the formation and decay of a complex were first proposed in the 1950s to describe nuclear collisions [3]. These models were then applied to molecular collisions [4]. Molecular statistical theories were put on a firm theoretical footing by Miller [5], who used as justification the formal theory of resonant collisions [6–8].

Pechukas and Light [9, 10] pioneered a statistical theory to predict the rate and product internal state distribution of the reaction of an atom with a diatomic molecule. This theory was based on Light's work on the phase space theory of chemical kinetics [11, 12] but, in addition, imposed detailed balance. This work has formed the basis of modern quantum mechanical treatments of complex-forming chemical reactions [13–17]. Here, once the complex is formed, it can fall apart to yield any accessible reactant or product subject to conservation of the total energy and angular momentum.

Pechukas and Light [9, 10] made some additional simplifications: First, they assumed that the capture probability was zero or one, depending on whether the reactants had sufficient energy to surmount the centrifugal barrier for each partial wave (related to the total angular momentum J of the collision complex). Second, they assumed that the long-range potential could be described by an inverse power law, – c01-math-006 . These assumptions, particularly the latter, were reasonable in an era where calculation of a potential energy surface (PES) was a major undertaking.

Subsequently, Clary and Henshaw [13] showed how to apply time-independent (TID) coupled-states and close-coupling methods to the determination of capture probabilities for systems with anisotropic long-range interactions. More recently, Manolopoulos and coworkers [14, 15] combined the statistical considerations of Pechukas and Light with the Clary–Henshaw TID quantum capture probabilities, using, in addition, accurate ab initio potential energy surfaces. This so-called quantum statistical method has been applied to a number of atom–diatom reactions that proceed through the formation and decay of a deeply bound complex. Guo has demonstrated how a time-dependent (wavepacket, WP) determination of the scattering wave function can be used in an equivalent quantum-statistical investigation of reactions proceeding through deep wells [18].

In related work, Quack and Troe [19] developed an adiabatic channel model to describe the unimolecular decay of activated complexes. This theory has been applied to a variety of processes, including the OH + O reaction [20].

González-Lezana [21] has written a comprehensive review of the use and applicability of quantum statistical models to treat atom–diatom insertion reactions. A good agreement with full quantum reactive scattering calculations has been found for properties such as the differential cross sections for the reactions of C( c01-math-007 D) and S( c01-math-008 D) with c01-math-009 , while less satisfactory agreement was found for the O( c01-math-010 D) and N( c01-math-011 D) + c01-math-012 reactions [15]. This comparison illustrates a limitation of the statistical theory: how to assess the accuracy of the approach without recourse to more onerous calculations. The differential cross-section of the product of a statistical reaction should have forward–backward symmetry. This is often not quite the case because the quenching of interferences between partial waves is not complete, particularly in the forward and backward directions [22].

Typically, fully quantum scattering calculations involve expansion of the scattering wavefunction in terms of all the triatomic states that are energetically accessible during the collision. The computational difficulty scales poorly with the number of these internal states. Both the large number of accessible vibrational states of a triatomic as well as the rotational degeneracy of the states corresponding to the A + BC orbital motion contribute to this bottleneck. Deep wells in any transient c01-math-013 complexes are particularly problematic. An example is the O( c01-math-014 D) + c01-math-015 c01-math-016 OH + H reaction, for which the PESs are illustrated schematically in Fig. 1. Even without taking anharmonicity into account, there are >1900 c01-math-017 O vibrational levels with energy below the O( c01-math-018 D)+ c01-math-019 asymptote. And this does not include rotational levels. Thus, full quantum reactive scattering calculations for complex-mediated reactions are a heroic task [23].

Schematic diagram of the potential energy surfaces of the OHH system.

Figure 1 Schematic diagram of the potential energy surfaces of the OHH system. Only the lowest ( c01-math-020 ) PES was taken into account in the initial quantum statistical calculations on the O( c01-math-021 D) + c01-math-022 reaction [14, 15]. Note that c01-math-023 O ( c01-math-024 ) well lies c01-math-025 59,000 c01-math-026 below the O( c01-math-027 D) + c01-math-028 asymptote.

Adapted from Rackham et al. 2001 [14] and Rackham et al. 2003 [15].

In the quantum statistical method, the close-coupled scattering equations are solved outside of a minimum approach distance, the capture radius c01-math-029 , at which point the number of energetically accessible states (open channels) is much less than at the minimum of the c01-math-030 complex. Also, because the point of capture occurs well out in the reactant and/or product arrangement, one does not need to consider the mathematical and coding complexities associated with the transformation from the reactant to product states [24]. It is these simplifications that make the quantum statistical method so attractive.

Formation of a transient complex does not always lead to chemical reaction. The complex may decay to the reactant arrangement, resulting in an inelastic collision. The present review outlines the application of quantum statistical theory to nominally nonreactive collisions that access PESs having one or more deep wells. Consider, the generic A + BC c01-math-031 c01-math-032 c01-math-033 AB + C collision. The inelastic event A + BC(v, c01-math-034 ) c01-math-035 A + BC( c01-math-036 ) can occur in a direct (non-complex-forming) collision, either through an encounter in which the partners approach in a repulsive geometry or at a larger impact parameter, where the centrifugal barrier prevents access to the complex. In addition, complex formation ( c01-math-037 ) and subsequent decay into the reactant arrangement will also contribute to inelasticity.

In general, weak, glancing collisions contribute substantially to rotational inelasticity. Thus, one might naively expect that both direct and complex-forming processes will contribute to rotational inelasticity. By contrast, vibrational inelasticity in collisions on basically repulsive PESs is very inefficient [25, 26], because the variation of repulsive PESs with the vibrational modes of the molecular moiety is weak. Thus, one might anticipate that the formation of a transient complex, in which substantial change in the bond distances might occur, could make a major contribution to vibrational relaxation.

Also, for A + BC collision systems where one of the reactants is an open-shell species, typically (as shown schematically in Fig. 1), there are a number of electronic states that correlate with the A + BC (or AB + C) asymptote. Of these states, one (or, only a few) leads to strongly bound c01-math-038 intermediates, while the others are repulsive. The branching between the energetically accessible fine-structure levels of the products (in the case of OH, the spin–orbit and c01-math-039 -doublet levels) will be controlled by the coupling between the various electronic states as they coalesce in the product arrangement, as the complex decays. We might predict that this branching, which can often be measured experimentally [27, 28], would be insensitive to any couplings within the c01-math-040 complex, where the excited electronic states lie high in energy, and hence be an ideal candidate for prediction by a quantum-statistical calculation.

Reactions involving isotopologs of the same atom,

1

equation

where c01-math-042 designates an isotopolog of B, are an additional example where the quantum statistical method can provide predictions and useful insight into potential experiments.

The next section contains a formal review of the quantum-statistical method, followed, in the remainder of this review, by a discussion of applications of this method to the problems introduced earlier in this section. In addition, and related to our discussion of inelastic collisions in the presence of a collision complex, we will use the quantum statistical method to calculate transport cross sections, which are weighted averages of differential cross sections. Here, the goal will be the determination of transport cross sections for the A + BC collision pair, in the presence of a deep c01-math-043 well.

II. Quantum Statistical Theory

Here, we describe the extension of the quantum statistical method to inelastic scattering, in the TID formulation due originally to Manolopoulos and coworkers [14, 15, 21]. Guo and coworkers have described an equivalent time-dependent formulation [29, 30], which they have applied to a number of reactive collisions. In principle, this time-dependent methodology could also be applied to inelastic scattering.

Consider the collision of two particles with internal structure, for example an open-shell molecule or atom, with total angular momenta c01-math-044 and c01-math-045 , respectively. We suppress any other labels, for example the fine-structure manifold c01-math-046 for an open-shell molecule, required to designate fully the levels. The integral cross section for a transition between the initial level pair c01-math-047 = ( c01-math-048 , c01-math-049 ) and a final level pair c01-math-050 = ( c01-math-051 , c01-math-052 ) at total energy E is given by the expression

2

equation

In Eq. (2), c01-math-054 is the internal energy of the initial level pair, c01-math-055 is the collision reduced mass, J is the total angular momentum, c01-math-056 and c01-math-057 are the initial and final orbital angular momenta, and c01-math-058 . The angular momenta c01-math-059 and c01-math-060 are vector sums of c01-math-061 , and c01-math-062 , respectively. Note that the cross section for transition from pair c01-math-063 to pair c01-math-064 involves, implicitly, a multiple summation over the projection quantum numbers of both angular momenta, as well as that of the orbital angular momentum of the collision partners. For the collision of a molecule with a structureless atom, we have c01-math-065 , c01-math-066 , and c01-math-067 .

The thermal rate constant as a function of temperature is given by [31]:

3

equation

In Eq. (3), c01-math-069 is the Boltzmann constant and c01-math-070 is the collision energy ( c01-math-071 = E – E c01-math-072 ).

In a quantum description, the probability c01-math-073 of a transition between the initial and final scattering states (channels) for total angular momentum c01-math-074 is given by the square modulus of the c01-math-075 (or c01-math-076 ) matrix element between these states:

4

equation

The c01-math-078 matrix can be obtained by the imposition of scattering boundary conditions with a TID close-coupling determination of the scattering wave function. Several extensive reviews of the general equations for inelastic scattering (with application to rotationally inelastic scattering) are available [32–34].

In the quantum statistical theory [14, 15], the probability c01-math-079 in Eq. (2) is computed as

5 equation

where c01-math-081 is the capture probability, namely the probability of forming the collision complex from the initial level pair c01-math-082 in the scattering channel c01-math-083 for total angular momentum c01-math-084 . In Eq. (5), c01-math-085 is the fraction of collision complexes with total angular momentum c01-math-086 , which dissociates into the final level pair in the scattering channel c01-math-087 and equals

6 equation

As discussed by Rackham et al. [14], the quantum statistical cross sections obey detailed balance.

Since the scattering event can lead to the formation of a collision complex, the c01-math-089 matrix element is not unitary. Unlike the theory of Pechukas and Light [9, 10], the quantum statistical capture probability can lie between zero and one. The capture probability for initial level pair c01-math-090 in the scattering channel c01-math-091 can be computed as

7

equation

where the sum is over all open channels.

The scattering equations for the quantum statistical theory are identical to those in conventional inelastic scattering. The difference lies in the boundary conditions. In the quantum statistical theory, the close-coupling equations are integrated out from the capture radius, which defines the outer extent of the complex. As in the treatment of inelastic and reactive collisions, it is convenient to use the log-derivative method [35, 36]. In our work, we have implemented Airy boundary conditions, corresponding to a linear reference potential at c01-math-093 [37]. In this way, for systems involving multiple attractive and repulsive PESs at c01-math-094 , fluxes on all the surfaces are treated on an equal footing. Unlike conventional scattering, for which the log-derivative matrix is real, here the log-derivative matrix is complex. The radial scattering differential equations are integrated using a linear reference potential in each sector [38].

An alternative to solution of the full, close-coupling equations is the use of the coupled-state approximation (CS) [39]. Rackham et al. [14] provided details on the implementation of the CS approximation within the quantum statistical method.

In applications of the quantum statistical theory, the scattering wave function, and hence its logarithmic derivative, is usually expanded in a body-fixed basis since the interaction is expressed more simply than in a space-fixed basis. In the asymptotic region, the log derivative is transformed to the space-fixed frame [40], and the standard scattering boundary conditions are applied.

In reactive collisions, the product can be formed only by formation and decay of the collision complex, that is, in an indirect collision. By contrast, collision-induced inelastic transitions can proceed both directly, without the intervention of a collision complex, and indirectly. The probability c01-math-095 in the expression in Eq. (2) for the state-to-state cross section is given by the exact expression (Eq. (4)) for a direct collision added to the cross section predicted by the statistical approximation (Eq. (5)) for an indirect collision.

III. Fine-Structure Branching in Reactive O(¹D) + H2 Dynamics

In the original application of the quantum statistical method to the O( c01-math-096 D) + c01-math-097 c01-math-098 OH + H reaction [14, 15], a single PES, of c01-math-099 symmetry and including the c01-math-100 O c01-math-101 well, was employed. As discussed above, these calculations were compared with full, quantum reactive scattering calculations [23] to check the validity of the statistical approximation.

This single-PES treatment of this reaction does not capture several important features of the dynamics. As Fig. 1 shows, there are four PESs that emanate from the product OH + H asymptote. The two states of triplet spin multiplicity correlate with the O( c01-math-102 ) + c01-math-103 asymptote [41]. The highest-energy state, c01-math-104 , correlates with the first excited singlet c01-math-105 O state [42]. The energy spacing between these PESs decreases as the products separate, and we may expect nonadiabatic effects in the dynamics.

In addition, the OH molecule is an open-shell radical possessing nonzero orbital and spin angular momenta. As shown in Fig. 2, the levels of OH are split into two fine-structure manifolds separated by the spin–orbit splitting. The lower/upper fine-structure manifolds are designated c01-math-106 and c01-math-107 , respectively. The lower rotational levels lie close to the Hund's case (a) limit, with projection quantum numbers c01-math-108 and c01-math-109 , respectively, for the two manifolds, but rapidly go to intermediate-case coupling as c01-math-110 increases. Each rotational/fine-structure level c01-math-111 c01-math-112 , where c01-math-113 or 2, is further split into two nearly degenerate levels, called c01-math-114 -doublets, of opposite parity, with symmetry index c01-math-115 equal to c01-math-116 and c01-math-117 for the c01-math-118 and c01-math-119 levels, respectively [43, 44].

In their experimental investigations of the dynamics of the O( c01-math-120 D) + c01-math-121 reaction, Butler, Wiesenfeld, and their coworkers [27, 28] employed laser fluorescence detection to probe the OH products over a wide range of rotational levels. The most striking feature of the product state distribution was the propensity for the OH product to appear preferentially in the c01-math-122 c01-math-123 -doublet levels [45]. This aspect of the product state distribution is not captured in the single-PES calculations.

Graphical illustration of energies of the lower rotational/fine-structure levels of the OH(X2π, υ = 0) manifold. The Λ-doublet splitting has been exaggerated for clarity.

Figure 2 Energies of the lower rotational/fine-structure levels of the OH( c01-math-124 , c01-math-125 ) manifold. The c01-math-126 -doublet splitting has been exaggerated for clarity.

By including all four OHH PESs (Fig. 1) and the open-shell nature of the OH product, Alexander et al. [37] were able to compute multiplet-resolved cross sections for the formation of the OH rotational/fine-structure product levels. They carried out internally contracted multireference configuration interaction (icMRCI) calculations [46] of the four PESs emanating from the OH + H asymptote as a function of the OH bond length c01-math-127 and the Jacobi coordinates c01-math-128 and c01-math-129 . Figure 3 presents plots of these PESs for the OH bond length fixed at the equilibrium value c01-math-130 . We see that the c01-math-131 PES is quite anisotropic, being significantly repulsive near linear geometry, but is strongly attractive in bent geometries, leading to the deep c01-math-132 O( c01-math-133 ) minimum. The other three PESs are considerably less anisotropic and primarily repulsive (except for a weak van der Waals attractive region in a long range). We see that the c01-math-134 and c01-math-135 states of a given spin multiplicity are degenerate in linear geometry; the linear approach of OH( c01-math-136 ) to H( c01-math-137 ) yields c01-math-138 and c01-math-139 states, both doubly degenerate.

Graphical illustration of contour plots (in cm-1) of the OH(r0)-H PESs for the (top row) 1A', 3A',and (bottom row) 1A.

Figure 3 Contour plots (in c01-math-140 ) of the OH( c01-math-141 )−H PESs for the (top row) c01-math-142 , c01-math-143 , and (bottom row) c01-math-144 , c01-math-145 states determined at the c01-math-146 vibrationally averaged OH bond length c01-math-147 . The angle c01-math-148 corresponds to linear OHH geometry.

As will be seen in the next sections, the availability of these PESs allows calculation of cross sections and rate constants for OH + H inelastic collisions, as well as treatment of the O( c01-math-149 D) + c01-math-150 reaction. The treatment of the scattering with inclusion of all four PESs and the open-shell nature of the OH and H collision partners is considerably more complex than in the single-PES calculations.

IV. Inelastic OH + H Collisions

Atahan and Alexander [47] reported a theoretical study of rovibrational relaxation in OH + H collisions. For these processes, relaxation can occur through direct (noncapture) scattering, without exchange of H, namely

8

equation

by decay of the complex back to the original arrangement

9

equation

or by decay of the complex accompanied by H exchange

10

equation

Schematically, the PESs can be adapted from Fig. 1, as shown in Fig. 4.

Schematic diagram of the potential energy surfaces relevant to OH(υ, j) +H → OH(υ', j') + H inelastic scattering.

Figure 4 Schematic diagram of the potential energy surfaces relevant to OH( c01-math-154 , c01-math-155 ) + H c01-math-156 OH( c01-math-157 ) + H inelastic scattering. Relaxation can occur through both noncapture (direct) inelastic scattering (Eq. (8)) as well as direct (no hydrogen exchange; Eq. (9)) and exchange processes (Eq. (10)), both of which can sample the deep c01-math-158 O well.

Here, we outline the treatment of the inelastic scattering dynamics, including the electronic degrees of freedom. The total Hamiltonian for the OH + H system can be written as

11

equation

In Eq. (11), the electronic coordinates are denoted collectively as q. The total Hamiltonian includes the nuclear kinetic energy c01-math-160 , the electronic interaction c01-math-161 between OH and H, and the Hamiltonian c01-math-162 describing the isolated OH molecule. The latter includes vibrational and rotational motion of OH, as well as the spin–orbit interaction and the c01-math-163 doubling [48].

The overall wave function of the OH–H system was expanded in an uncoupled, Hund's case (a) basis

12

equation

Here, the c01-math-165 and c01-math-166 designate full and reduced rotation matrix elements [49], c01-math-167 denotes the Euler angles relating the space and body frames, and c01-math-168 is the total angular momentum, with projections c01-math-169 and c01-math-170 along the space-frame c01-math-171 and body-frame c01-math-172 (i.e., along c01-math-173 axis), respectively. The function c01-math-174 describes the OH vibrational wave function. The rotational angular momentum of the OH radical is c01-math-175 , which has projections c01-math-176 and c01-math-177 along c01-math-178 and c01-math-179 , respectively. The ket c01-math-180 represents the OH electronic wave function, where c01-math-181 and c01-math-182 are the projections of the OH electronic orbital and spin angular momenta along c01-math-183 and c01-math-184 . The ket c01-math-185 denotes the electronic wave function of the H atom, with projection c01-math-186 of the H atom spin along c01-math-187 .

The matrix elements of the electronic Hamiltonian c01-math-188 in the basis defined in Eq. (12) are linear combinations of the interaction potentials of the four OHH electronic states. The calculation of these matrix elements is described in detail in Ref. [37]. It should be noted that the matrix elements were determined by numerical quadrature in c01-math-189 and c01-math-190 . The rovibrational/fine-structure energies and wave functions of the OH radical are obtained by diagonalizing c01-math-191 in a definite-parity, Hund's case (a) basis defined by

13

equation

where c01-math-193 can take on the positive-definite values c01-math-194 and c01-math-195 , and c01-math-196 . The c01-math-197 and c01-math-198 c01-math-199 -doublets have c01-math-200 and c01-math-201 , respectively. In this basis, the matrix of c01-math-202 is diagonal in c01-math-203 , and its matrix elements have been given previously [48, 49].

The OHH wave function is expanded in the basis of Eq. (12). Premultiplication by each of the basis functions, integration over all electronic and nuclear coordinates except c01-math-204 , and evaluation of the resulting matrix elements lead to the corresponding set of close-coupled equations. These equations are solved subject to the capture boundary conditions, as discussed in Section II.

For direct scattering, the inelastic transition probabilities and resulting cross sections are proportional to the square of the corresponding c01-math-205 -matrix elements [Eq. (4)]. For the contribution to the inelastic cross sections arising from collisions that enter the attractive well, the cross sections are given by the quantum statistical expression [Eq. (5)] regardless of whether the identity of the proton is retained or not. Further, since the initial and final H atom spin projection is not specified, cross sections between specific OH initial and final rotational/fine-structure levels are computed by averaging and summing over the initial and final H atom spin projections:

14

equation

The following sections describe the application of the quantum statistical method for the calculation of cross sections and rate constants for various processes involving collisions of the OH radical with hydrogen atoms. These calculations have employed the above formalism with the PESs presented in Section III.

A. OH + H Vibrational Relaxation

The OH radical is an important species in combustion, the earth's atmosphere, and in the interstellar medium. In the earth's atmosphere, vibrationally excited OH is generated in the mesopause and the stratosphere and troposphere, by reactions 15 and 16, respectively:

15 equation

16 equation

Knowledge of the rates of OH vibrational relaxation is needed to model the atmospheric chemistry of the OH radical, which is an important oxidant in the atmosphere. The collisional vibrational relaxation of OH( c01-math-209 ) by inert gasses and diatomic and polyatomic molecules has been the subject of a number of experimental studies (see [50] and references cited therein).

Smith and Williams [51] have argued that the rate of vibrational relaxation for interaction of potentially reactive collision partners should be greater than for nonreactive encounters. In particular, in a radical–radical collision, a deep well can usually be accessed without a barrier. In this case, as discussed in Section I., the bond length of the molecular collision partner(s) should be strongly coupled with the other degrees of freedom. Consequently, a statistical model will be appropriate to compute the rate of vibrational relaxation. A further consequence is that the relaxation rate should not depend significantly on the vibrational level since the rate of relaxation will depend primarily on the rate of formation of the collision complex [51].

We have seen in Figs. 3 and 4 that a deep well exists in the PES of one of the states emanating from OH + H. Thus, the quantum statistical method [14, 15] should be appropriate to compute the rate of OH vibrational relaxation in collision with H atoms. Atahan and Alexander [47] employed the coupled-states [14, 39] version of the quantum statistical method to compute the rate of vibrational relaxation of OH( c01-math-210 , 2) in collision with H atoms. As discussed earlier (Eqs. 10), there will be both direct and indirect contributions to the cross section, and hence the rate constant (Eq. (3)). Recall (Fig. 1) that the O( c01-math-211 D) + c01-math-212 channel is endoergic, so that the OHH complex can only decay to OH + H.

Figure 5 displays the dependence on the collision energy of the cross section for vibrational relaxation of the lowest OH( c01-math-213 ) rotational/fine-structure level ( c01-math-214 c01-math-215 , see Fig. 2) to the c01-math-216 ground vibrational level, summed over all final rotational/fine-structure levels. We see that the vibrational relaxation cross section is dominated by the contribution from formation and decay of the OHH collision complex. The indirect contribution decreases with increasing collision energy. The inset in Fig. 5 shows that the direct contribution increases with increasing collision energy, but is insignificant compared to the indirect contribution for all collision energies. We observe, as argued by Smith and Williams [51],