Theories and simulations of roaming

Joel M. Bowman *a and Paul L. Houston *bc
aDepartment of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University Atlanta, Georgia 30322, USA. E-mail: jmbowma@emory.edu; Tel: +1-404-727-659
bDepartment of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14850, USA. E-mail: PLH2@cornell.edu
cSchool of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332-0400, USA

Received 7th August 2017

First published on 5th October 2017


The phenomenon of roaming in chemical reactions has now become both commonly observed in experiment and extensively supported by theory and simulations. Roaming occurs in highly-excited molecules when the trajectories of atomic motion often bypass the minimum energy pathway and produce reaction in unexpected ways from unlikely geometries. The prototypical example is the unimolecular dissociation of formaldehyde (H2CO), in which the “normal” reaction proceeds through a tight transition state to yield H2 + CO but for which a high fraction of dissociations take place via a “roaming” mechanism in which one H atom moves far from the HCO, almost to dissociation, and then returns to abstract the second H atom. We review below the theories and simulations that have recently been developed to address and understand this new reaction phenomenon.


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Joel M. Bowman

Joel M. Bowman is Samuel Candler Dobbs Professor of Chemistry at Emory University. He has been an Alfred P. Sloan Research Fellow and is a Fellow of the American Physical Society, a member of the International Academy of Quantum Molecular Sciences and recipient of the Dynamics of Molecular Collisions Herschbach Medal in Theory.

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Paul Houston

Paul L. Houston is Peter J. W. Debye Professor of Chemistry Emeritus at Cornell University and Professor Emeritus of Chemistry and Biochemistry at Georgia Institute of Technology. He has been an Alfred P. Sloan Research Fellow (1979–1981), a Camille and Henry Dreyfus Teacher Scholar (1980), and a John Simon Guggenheim Fellow (1986–1987). In 2001 he shared with David W. Chandler the Herbert P. Broida Prize of the American Physical Society for work on product imaging in chemical dynamics. He was elected a fellow of the American Academy of Arts and Sciences in 2003.


1 Introduction

Chemistry is often defined to include the identification of substances, their structures and their properties; the determination of the processes by which they interact, combine and change; and the use of these processes to form new substances. Of paramount practical importance is the estimation of how fast a particular interaction, combination or change will occur. Based on quantum and statistical mechanics and used for nearly a century, transition-state theory has been the foundational method for predicting the rates of chemical processes. For unimolecular processes, transition-state theory takes the form of the RRKM (Rice–Ramsperger–Kassel–Marcus) expression, which usually provides an accurate prediction of the microcanonical rate constant. The rate is proportional to the ratio between the number of states that can lead to products at the energy above the geometry of a “transition state” and the product of Planck's constant and the density of states of the reactant at the same total excitation energy. The transition state (TS), located on a dividing surface between reactants and products, is usually taken to be the highest energy point on the minimum-energy path between reactants and products, although a more general approach defines it as the point on the path where the reactive flux is a minimum.

In recent years however, conventional transition-state theory has been called into question by the discovery of a number of reaction systems, found both experimentally and theoretically, that stray very far from the minimum-energy path. For example, “roaming” describes trajectories that enter the long-range, high-energy part of the potential that is normally associated with dissociation to radicals, but which return to the region of the potential that leads to molecular products. The question of how to deal with reactions that can take more than one path to products is entwined with the question of how to determine the nature of a surface dividing one pathway from the other. While theory has debated such questions, experiments have provided more and more examples. Roaming has now been observed in formaldehyde,1–16 CH3CHO,17–24 larger aldehydes,25 acetone,15 alkanes,26 methyl formate,27–31 NO3,15,32–35 methyl nitrite,36 CH3NO2,36–38 CO2,39 Criegee intermediates,40 2-hydroxypyridine,41 and in the recent time-resolved studies of H3+ formation.42 Roaming in bimolecular reactions has also been indicated.43–49

Fig. 1 shows a schematic of how roaming occurs in H2CO and NO3. For H2CO a 3D perspective plot of the potential is given indicating the deep H2CO well and the simple bond fission pathway to the radical channel H + HCO. Superimposed is a schematic of the roaming pathway as a “stage-right turn” away from that pathway and instead moving to a flat region of the potential that eventually leads to a “self-abstraction” of the incipient H⋯HCO fragments to give H2 + CO. The lower plot depicts the dissociation of NO3 in the ground electronic state, which like H2CO begins to dissociate to the radical products O + NO2, but instead of separating to those products the incipient fragments roam to an orientation where O-atom abstraction occurs to form O2 + NO. In this plot the potential energy is also plotted and, as seen, it reaches a high value and is nearly constant during the time that roaming occurs. The large oscillations in the potential in the final segment of the plot indicate highly excited motion in the fragments. It turns out that most of the excitation is in the O2 vibration.34


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Fig. 1 Schematic of roaming in H2CO and NO3.

Since the term “roaming” was coined in 2004, there have been a number of reviews of roaming.50–56 With the exception of the most recent one,56 these have presented reviews of experimental and computational, mainly quasi-classical trajectory calculations (QCT), studies of roaming. The present review is focused on models of roaming as well as new QCT calculations. These models and calculations have used roaming in H2CO as the example of primary importance, in part because this was the first system where roaming was observed. A second important aspect of the dynamics of H2CO is the existence of the conventional tight transition state pathway to the molecular products as well as the roaming pathway. Finally, this is a relatively small system where roaming occurs and full-dimensional, potential energy surfaces (PESs) describing these pathways to the molecular products exist.11,57

2 Theories and simulations

In the review that follows, we will comment on a variety of theoretical and simulation methods applied to roaming. These include quasi-classical trajectory studies, the roaming saddle point and intrinsic reaction coordinates, kinetic/statistical theories, phase space theories, geodesics: pathways connecting the initial configuration to roaming, and quantum mechanical treatments of roaming, both dynamical and spectroscopic.

2.1 Quasi-classical trajectory studies of roaming

Quasi-classical trajectory studies have provided important information about the mechanism of roaming. In addition to formaldehyde, the MgH2 system has been studied using both QCT and quantum wave packets,44 as will be discussed in Section 2.6; acetaldehyde has been studied by Shepler, Braams, and Bowman19 and by Haezlewood et al.;20 nitromethane and methyl nitrite have been investigated by Dey et al.36 and Homayoon et al.;37 and NO3 has been investigated experimentally and theoretically by several groups.15,32–35,58–60 Nonetheless, most of the previous work has been performed on formaldehyde5,7–10,12,13,61–63 and the related H + HCO reaction.16,64 It is also worth noting that classical impulsive models have also been applied to formaldehyde.25,29

Perhaps the most detailed QCT study of formaldehyde is a recent report, in which a statistical analysis of trajectories provided insight into the roaming process.65 Nine excitation energies from 34[thin space (1/6-em)]500 to 41[thin space (1/6-em)]010 cm−1 above the global minimum were investigated. Trajectories were run using the 2004 potential energy surface of Zhang et al.,11 which is a fit to 80[thin space (1/6-em)]000 CCSD(T) ab initio energies and 50[thin space (1/6-em)]000 MRCI/aug-cc-PVTZ energies. Integration of the trajectories was performed with 0.05 fs integration steps and with atomic locations and momenta recorded every 6.25 fs. Reaction to H2 + CO, especially through the roaming channel, is essentially a threshold process, so the authors paid careful attention to zero-point energy violations. The trajectories were projected onto pairs of three polyspherical66 coordinates: rH, the distance between the CO center of mass to the furthest H atom, and the azimuthal (ϕ) and polar coordinates (θ) of that H atom with respect to the CO axis. The coordinate ϕ is measured from the plane of the remaining HCO and shown for values only from 0 to π since the positions and energies for positive and negative values are the same. Fig. 2 shows a typical trajectory for a reaction that takes place through the traditional transition state projected onto the rHϕ plane. The formaldehyde equilibrium position (the global minimum) is at ϕ = π and the transition state is near ϕ = 1. Reaction to H2 + CO occurs near the asterisk.


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Fig. 2 H2CO → H2 + CO trajectory that proceeds through the traditional transition state. The asterisk marks the point of reaction. The total energy for this trajectory is 36[thin space (1/6-em)]000 cm−1 above the global minimum.

In contrast, Fig. 3 shows a roaming trajectory. The rH coordinate extends substantially as ϕ changes, so that by the time the “roaming” H atom reaches ϕ = 0, the location of the “non-roaming” H atom, its rH coordinate is too large for reaction. The straight lines indicate a nearly uniform velocity in ϕ; the roaming H atom rotates around the CO axis as rH slowly extends and then contracts. Just before the reaction, rH has gotten small enough so that the two H atoms can interact. Reaction takes place at the location marked by the asterisk.


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Fig. 3 H2CO → H2 + CO trajectory that proceeds through roaming. The asterisk marks the point of reaction. The total energy for this trajectory is 41[thin space (1/6-em)]010 cm−1 above the global minimum.

The authors found that such azimuthal rotation of the roaming H atom was the principal motion involved in roaming reactions. The azimuthal rotation is nearly uniform and is sometimes accompanied by rotation in the polar coordinate. Other results from the trajectory study provided (a) flux maps, showing that the reactive flux for roaming reactions occurs at higher rH than for transition state reactions, (b) reaction configuration plots, showing that roaming reactions occur from a configuration that has a considerably elongated rH coordinate, (c) average numerical results and distributions measuring 46 different properties such as reaction time and CO vibrational energy for roaming and transition state trajectories, (d) correlation plots showing, for example, the correlation of the internal energy of the H2 with that of the CO, and (e) vector correlation results, showing, for example, the anisotropy, rotational alignment, and vJ correlation for CO and H2. Some typical results are that, depending on excitation energy, the fraction of trajectories that roam is 2.6–15%, the fraction of trajectories that produce H + HCO varies from 0–58%, for those that roam their average time spent roaming is 0.32–0.07 ps, and, for all trajectories, the dissociation time ranges from 56–0.7 ps. The vibrational energy distribution for H2 roaming trajectories is much more highly excited than that for transition-state trajectories. For example, at an excitation energy of 36[thin space (1/6-em)]223 cm−1 the average H2 vibration energy is 21[thin space (1/6-em)]600 cm−1 for roaming trajectories and only 7530 cm−1 for transition-state ones. Conversely, transition-state trajectories have much higher average rotational excitation (J = 41) than roaming ones (J = 20).

These observations are in excellent agreement with previous experimental results as shown both in this study65 using the 2004 potential energy surface11 as well as in a more recent study67 using a new potential energy surface.57 As an example, a comparison of the results of recent experiments and trajectories67 is shown in Fig. 4 for the CO speed distribution for selected CO rotational levels. The QCT results, which were obtained using standard and Gaussian binning of the final ro-vibrational states of CO and H2, are in excellent agreement with the new experiments. This level of agreement is even better than the good agreement seen earlier with experiment, using the 2004 PES.5


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Fig. 4 CO(v = 0, J) speed distributions for two different J values. The green/blue lines give the experimental/trajectory results for even parity (solid) or odd parity(dashed). The rotational levels in the panels are (a and b) J = 10 and (c and d) J = 40. The comb at the top of each panel gives the speed corresponding to the indicated vibrational levels of H2.

2.2 Roaming saddle point and intrinsic reaction coordinate

A “roaming transition state” and an associated intrinsic reaction coordinate (IRC) were reported for H2CO by Harding and co-workers.68 Actually, these authors reported a first-order saddle point (SP) in the region of roaming identified in quasi-classical trajectory calculations.5 Subsequent work casts doubt (see below) on the validity of this saddle point as the transition state, and a detailed critique of a roaming saddle point vis-à-vis a roaming transition state has been made in the context of the general phase space theory of roaming,56 discussed below. Nevertheless, in the absence of more sophisticated approaches, which generally require extensive knowledge of the potential energy surface, roaming SP(s) and the associated IRCs are useful indicators of roaming. Indeed, a prominent example of a roaming IRC that solved an important problem in photochemistry was reported by Morokuma and co-workers for the photodissociation of NO3 to NO + O2.15,32,33,58–60 This is because roaming pathways are the only pathways to these products and there are roaming pathways on both the ground and first excited electronic states. This work also importantly extended the idea of roaming to electronically excited states. These IRCs actually have several saddle points on them and so it is not clear which one is relevant for controlling the dynamics. Further, the degree of vibrational excitation of the O2 product is very different depending on the electronic state where roaming occurs. The roaming IRCs do not provide a means for making quantitative comparisons with experiment concerning these distributions. However, subsequent QCT calculations on full dimensional PESs were done for both electronic states and agree very well with experiment.34

A recent study applying an impulsive model at the roaming saddle point and along the IRC for H2CO, aimed at determining the final-state H2 and CO distributions, was reported by Tsai and Lin.25 They found that the results from the roaming SP were not in agreement with experiment and quasi-classical trajectory calculations. However, by applying this model on various positions on the IRC (past the SP and towards the molecular products) agreement with experiment and QCT improved. Unfortunately, as they noted, there appears to be no a priori way to determine this location on the IRC.

Another indication of the limitation to the roaming SP for obtaining quantitative predictions comes from Harding and co-workers,68 who applied standard RRKM theory to obtain the rate of the roaming reaction. The rate obtained at 1000 K, is roughly twenty times larger than the rate from the tight TS and clearly wrong. The interpretation of this failure of theory is that there is large re-crossing of the TS.

The QCT calculations described above for H2CO, as well as other calculations on roaming systems reviewed recently,55 clearly show the classical roaming flux deviates significantly from the roaming SP and IRC. This deviation has stimulated more sophisticated, non-dynamical approaches to roaming, as described next.

Before describing these non-dynamical approaches, we note that the photodissociation of H2CO begins on an electronically excited state (S1) which by a fairly complex, but understood process,34,60 populates the ground electronic state, where roaming occurs.

2.3 Kinetic/statistical theories of roaming

Kinetic or statistical models for roaming have been reported in the literature, as reviewed previously.55 Although we will concentrate here on applications to formaldehyde, it is important to recognize that there have been contributions to other roaming systems as well. For example, Harding, Georgievskii and Klippenstein have applied a novel approach to roaming in acetaldehyde,21 and Harding, Klippenstein and Jasper have considered how to determine dividing structures for use in statistical theories for roaming in MgH2, NCN, acetaldehyde, formaldehyde, and HNNOH.69 Finally, Ulusoy, Stanton and Hernandez have proposed a model for roaming in ketene.70–72 Subsequent study of ketene by Mauguière et al.73 used the phase space approach to be discussed in Section 2.4.

Klippenstein, Georgievskii and Harding described a five-state kinetic model in 2011, which was applied not only to formaldehyde but also to roaming in acetaldehyde, CH3OOH and CH3CCH.74 In the formaldehyde case the model had five species including two intermediates: formaldehyde, molecular products, radical products, a weakly interacting intermediate involving a partially dissociated CH bond, and another weakly interacting intermediate involving the two H atoms. These species are connected by seven rate constants. The steady-state assumption was used to determine branching between roaming and radical channels in various limits, and the best branching ratio results were found to be within a factor of two of trajectory calculations.

A study of the reactive fluxes as a function of energy predicted that at the lowest energy studied (0.1 kcal mol−1 above the H + HCO asymptote) the minima in the flux for the inner and outer transition states clearly showed substantially more flux through the inner transition state than the outer one, by approximately a factor of four. As the energy increased, the ratio of fluxes decreased, and the positions of the minima in the fluxes shifted to lower values of C–H separation. At very high energy, the distinction between the two channels completely disappeared. At low energies the prediction of the roaming to TS fraction was somewhat lower than the predictions of trajectories, whereas at higher energies it was somewhat higher. Overall, the agreement was quite reasonable.

Andrews, Kable and Jordan have also proposed a phase-space theory of roaming reactions.75 Their model depends on the difference in the roaming and radical threshold energies, where the former is about 161 cm−1 above the latter, and on Proam, a probability that those phase space states that may roam actually do so rather than recombining to regenerate H2CO. The authors assumed that most roaming states went on to products rather than recombining, so Proam was taken to be 0.99. The model predicted that the branching fraction for roaming decreased from about 1.0 to about 0.15 in the first 1000 cm−1 above the H + HCO threshold.

Houston et al.65 proposed a simple model that borrows some elements of each of the above approaches and provides a fit to the rate constant and branching data determined from trajectory studies. It considers the three species – formaldehyde, bimolecular products and radical products – but has only one intermediate, a roaming state, R, that may produce radicals, bimolecular products, or decay back to formaldehyde. The model is composed of five steps and their associated rate constants:

H2CO → R kR

R → H2CO k−R

H2CO → H2 + CO kTS-Bi

R → H + HCO kR-Rad

R → H2 + CO kR-Bi

R in these equations represents H–HCO in a roaming state, treated as if it were a highly fluxional isomer of formaldehyde. Because only a small fraction of trajectories return to the H2CO well, it is reasonable to neglect k−R compared to kR in the equilibrium that is set up by the first two processes (this is the same approximation as made by Andrews, Kable, and Jordan, above). The rate constant kTS-Bi is the rate constant for the normal transition state reaction giving the bimolecular products, H2 + CO. The total rate of H2CO dissociation determined by the trajectories gives the sum kR + kTS-Bi, whereas the average roaming time before reaction gives kR-Bi, the rate constant for formation of bimolecular products from R. The fraction of radicals and the fraction of roaming give two further equations, thus providing four equations with four unknown rate constants at each of the nine excitation energies considered by the authors. Solution provides the four rate constants. Plots of the log10 of the rate constants as a function of energy are shown in Fig. 5.


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Fig. 5 Plot of log10 of the rate constants (in ps−1) vs. excitation energy for the four rate constants describing a simple model for roaming. Blue is for kR-Rad, red is for kR-Bi, purple is for kR, and green is for kTS-Bi.

The general behavior of the rate constants as a function of energy seems quite sensible. At high energies, the rate constants have nearly parallel behavior with energy, indicating that there should be nearly constant product ratios as a function of energy, as found by the trajectories. At energies below about 37[thin space (1/6-em)]700 cm−1, the rate constants decrease rapidly near their respective thresholds. An RRKM calculation was performed that gave reasonable agreement with the energy dependence shown in Fig. 5. On the basis of these observations the authors argue that the roaming state should be treated as one might treat an isomer of formaldehyde. When entities are treated separately and densities of states are accounted for properly, it seems likely that transition state theory might still be used to calculate overall rate constants.

2.4 Phase space theories

An important approach to roaming comes from an appreciation of phase space theories. Mauguière et al.76 have pointed out that although the “energy landscape paradigm”, in which chemical transformations are viewed in terms of critical points on the potential energy surface, has served chemists well, it is actually phase space (momentum and position) rather than configuration space (position) that controls dynamics. The field of non-linear dynamics has shown that much of the richness in dynamical behavior cannot be inferred from the shape of the potential energy surface alone. Because roaming is essentially a dynamical process, important features of roaming must be considered from a phase space perspective. The importance of the saddle point in configuration space is supplanted in phase space by a so-called normally hyperbolic invariant manifold. This manifold is important for determining the dividing surfaces between dynamical pathways. In a manner similar to the dividing surfaces envisioned in transition state theory, the invariant manifolds have the properties of no local recrossing of trajectories and minimal flux.

A number of researchers have been active in applications of phase space theories to roaming processes. Ion–molecule reactions have been found to stray far from the minimum energy path,77–80 and Mauguière et al.45,46 have shown how these can be understood in terms of phase space theory. Other applications of the phase space approach have been to ketene,73 a system also studied by statistical models70–72 as mentioned in Section 2.3, ozone,76 MgH2,49 and formaldehyde.14

The work on formaldehyde was performed using a 2-degree-of-freedom reduced dimensionality system that included the distance from the center of mass of HCO to the roaming atom, R, and the angle θ between the CO axis and the vector R. Roaming was observed to occur along unstable periodic orbits with large excursions in θ. Fig. 6 displays the projections onto configuration space of the unstable periodic orbits, which are also the invariant manifolds for this two dimensional system. POTTS1 divides the reactant H2CO region from the region of the tight transition state complex, whereas POTTS2 divides the tight transition state complex from the reaction region leading to H2 + CO. POOTS is the dividing surface between a roaming complex and its dissociation to H + HCO; this surface lies along the centrifugal barrier to the radical products and defines an orbiting transition state. Finally, POROT separates the roaming and tight transition state regions. Many of the roaming trajectories follow a path near this structure, which, in phase space, shepherds the roaming trajectories toward reaction.


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Fig. 6 Periodic orbits in a reduced dimensional representation of roaming in formaldehyde. See text for an explanation. The energy units are in cm−1. (Reproduced from ref. 14 with permission from the American Chemical Society, copyright 2015).

The restriction to a 2D model may seem unrealistic, and in some ways it clearly is. However, the sophisticated non-linear phase space analysis undertaken by Mauguière et al. is already complex in 2D and it becomes significantly more complex in higher mathematical dimensions. This research group is likely the one that will make the leap into higher dimensionality to eventually provide a more realistic phase space characterization of roaming in H2CO.

2.5 Geodesics: pathways connecting the initial configuration to roaming

The goal of formulating a general dividing surface that serves as the transition state for roaming is clearly still a “work in progress”. A different perspective on roaming was recently proposed by Cofer-Shabica and Stratt.81 Here the goal was to correlate the region(s) of the initial configuration space with roaming. This was examined in detail for energized H2CO, using the PES of 2004 and what these authors term “geodesics”. These are pathways (not classical trajectories) in configuration space that connect the initial configuration space to roaming pathways. The geodesic pathways do not contain the oscillations associated with classical trajectories owing to the kinetic energy of atoms. In this respect they are superficially at least akin to reaction paths, with the distinction that the geodesic paths are short(est) length paths in Cartesian space connecting two configurations, without violating a total energy constraint. A roaming region was defined by two conditions, one is the inequality dmin < rHH < dmax, rHH is the distance between the two hydrogen atoms, and the second is that the restoring force between the distant H atom and the center of mass of the HCO fragment be “small” and less than an Fmax, which is negative. The values of the parameters were largely determined empirically by running thousands of classical trajectories.

The analysis of the geodesic pathways concludes that initial configurations with high potential values tend to lead to the roaming pathways and, more generally, the potentials along the roaming pathways are large. This finding is consistent with previous analyses of roaming dynamics where the potential was plotted vs. time and found to be much larger than the local kinetic energy.65

This geodesic analysis is an interesting step forward. However, the analysis does seem to require a global potential energy surface. This is not necessarily a criticism, since the goal of developing a theory that does not require a global potential energy surface to predict the existence and extent of roaming in a chemical reaction may not be realistic. Clearly such a goal is an “inheritance” of ordinary transition state theory, but roaming is anything but ordinary transition state theory.

2.6 Quantum theory of roaming – dynamical approach

Although there have been many experimental examples reported and several QCT studies of the dynamics of roaming, there has been only one quantum mechanical investigation of roaming, mainly because the examples of roaming systems are tetra-atomics and larger polyatomics which are still a major challenge for quantum calculations. We will consider two methods for finding the quantum mechanical signatures of roaming. In this section we consider the dynamics of the process and show an example of how wave packet calculations can identify this process in the triatomic H + MgH reaction. In the next section we consider how to identify spectroscopic features that can be used to determine the roaming wavefunction. Of course, these two methods are related, but they involve very different approaches to the problem.

Pioneering work on quantum roaming in the triatomic reaction H + MgH → Mg + H2 was reported by Takayanagi and Tanaka.82 In 2013 Li and Guo, re-examined this reaction both quantum mechanically and quasi-classically,44 using a new and more accurate PES.44,83 Very recently this PES was used in further QCT studies of roaming for D + MgD → Mg + D2 reaction.84

The H + MgH → Mg + H2 has several similarities to the H2CO unimolecular dissociation. It proceeds by two possible mechanisms. In one, the H atom and MgH molecule form an HMgH complex bound by about 73 kcal mol−1, which then can dissociate via a tight transition state to Mg + H2. The tight transition state is high, but below the MgH + H asymptotic limit by about 2.88 kcal mol−1. In the second mechanism, H simply abstracts the hydrogen atom from MgH in a barrier-less reaction. This situation raises the possibility, first suggested by Takayanagi and Tanaka,82 that after forming a complex, the H–MgH might almost dissociate to reactants but then react by the abstraction mechanism instead of crossing through the tight transition state. Indeed, Li and Guo find evidence for such roaming. As mentioned in Section 2.4, a phase-space interpretation of the reaction has recently been published by Mauguière et al.49

At low collision energies, it appears that the reaction is dominated by the direct abstraction channel, with the roaming channel responsible for only about 20 percent of the products. The tight transition state pathway is not at all appreciable at the collision energy of the study, 0.15 eV. As noted by Li and Guo, these results are consistent with earlier QCT studies of the bimolecular H + HCO reaction, which also occurs via these three pathways.16

The quantum mechanical calculations were performed using the Chebyshev real-wave packet method. The wavefunction was discretized in a mixed representation with direct-product discrete variable representations for the two radial coordinates and a finite basis representation for the angular coordinate. The authors also examined the reaction probability as a function of collision energy. This plot shows several resonance structures in what is a continuum of reaction probability. At the more prominent features, the authors investigated the wavefunction given by the wave packet at that energy. Some results are shown in Fig. 7 One can easily see large amplitude H excursions from the magnesium in the quasi-classical trajectory at the left. On the right, the close correspondence between the roaming classical trajectory and the roaming wavefunction confirms that the latter is indeed associated with roaming. Thus, the quantum mechanical signature of roaming is identified in this reaction as a large-amplitude vibrational feature appearing in energy just below the reaction threshold and persisting into the continuum.


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Fig. 7 The time evolution of a roaming trajectory in MgH2. On the left is the three-dimensional rendition of the QCT results. On the right is a two-dimensional contour plot showing the wavefunction (in blue), the potential energy (in black), and the projection of the QCT onto these coordinates (in red). The ordinate is the bending radians running from 0 to π; the abscissa is the distance of the incoming H atom from the MgH running from 2–15 a0. The black dots represent the positions of the two transition states. (Reproduced from ref. 44 with permission from the American Chemical Society, copyright 2013).

It is likely that these features of quantum roaming will be seen in larger systems, once the bottleneck to doing such calculations is overcome.

2.7 Quantum theory of roaming – spectroscopic approach

Dynamical calculations are not the only method for examining the quantum mechanical signatures of roaming; one can also examine the spectroscopy. The strong connection between the two has already been acknowledged above. Roaming is accompanied by resonance features, either vibrational eigenstates below the dissociation level or vibrational resonance features in the continuum above the dissociation level. Thus, another quantum approach is to try directly to determine the wavefunctions of these vibrational states. A further (speculative) goal, mentioned at the end of this subsection, is to use these states in a new interpretation of standard RRKM theory.

A convenient method for determining the wavefunction is provided by the MULTIMODE suite of programs.85 Briefly, MULTIMODE uses a potential energy surface to calculate the vibrational levels of molecules using the full Watson Hamiltonian in mass-scaled normal coordinates. The kinetic operators of the Hamiltonian are separable in the normal mode coordinates, but the potential energy will generally depend on all the normal mode coordinates. The next step of the program is to find the variationally best product wavefunction in a self consistent manner.86 The method is entirely analogous to a self-consistent field (SCF) electronic structure calculation. The next step, akin to including configuration interaction, is to take the resulting SCF functions as basis functions and to calculate the coupling between the vibrational modes. The coupling contributions are expanded into a series by considering all pairs, then all triplets, then all quartets, etc.87 The result is not only a set of final vibrational eigenstates but also, for each eigenstate of the system, a set of coefficients expressing that state as a weighted sum of the SCF basis states. Since the energies and wavefunctions of the SCF basis states are known, it is then possible to construct the wavefunction for the eigenstate.

A very useful property of the program is that the eigenfunctions can be determined starting from any stationary point on the potential energy surface, including both first- and second-order saddle points. This feature permits the examination of eigenfunctions that are very high in energy, including those above the dissociation limit. We use as an example the geometry of the “roaming” saddle point in formaldehyde. In this saddle point, the HCO is nearly in the equilibrium geometry of free HCO, while the roaming H atom is 7.4 Å from the center of mass of the CO. The energy of the saddle point is 32[thin space (1/6-em)]777 cm−1 above the global minimum and the zero-point energy at the saddle point is 12[thin space (1/6-em)]386 cm−1. For reference, the H + HCO asymptote energy is 33[thin space (1/6-em)]100 cm−1. The potential energy surface used for this configuration was that recently constructed by Wang, Houston, and Bowman.57

Fig. 8 shows the projection of the wavefunction for a selected excited state onto the rH and ϕ coordinates, where rH is the distance of the roaming H atom to the CO center of mass and ϕ is the difference in azimuthal angles between the two H atoms around the CO axis. The energy of the state is 13[thin space (1/6-em)]398 cm−1 above the zero point energy of the saddle point. The energy contours (red to orange) are assembled from trajectories and indicate the lowest energy of any trajectory passing through a particular rHϕ point. The wavefunction probability density (blue to red) shows several nodes in the ϕ coordinate, indicating almost free rotation. Most of the population is at geometries whose energy is above the high energy plateau, where roaming is observed to occur in quasi-classical trajectory calculations (for example, see Fig. 3).


image file: c7cs00578d-f8.tif
Fig. 8 Projection of the probability density of the wavefunction for an eigenstate 13[thin space (1/6-em)]398 cm−1 above the zero point energy of the second-order saddle point onto the rHϕ coordinates. The contour plot on which the wavefunction is superimposed shows the potential energy of the formaldehyde, as described in the text.

This calculation and examination of a delocalized roaming SP wavefunction confirms the extreme looseness of this SP region. Like the wavefunctions of the MgHH complex, this is still a suggestive and promising step in the right direction. However, the piece of puzzle that is still missing is to determine which of these wavefunctions actually represents roaming dynamics. That is, we need the know whether these wavefunctions (when suitably turned into fluxes) lead to the molecular products instead of say the radical ones. This, as already noted, is still a major challenge to do rigorously for systems larger than triatomics. Another goal, speculative at this point, is to develop a simple criterion, perhaps based on the coupling, encoded in these wavefunctions, to assign a given wavefunction as “roaming”. If this could be done, then we would propose a modification of RRKM for the microcanonical rate constant of roaming theory to read

 
image file: c7cs00578d-t1.tif(1)
where Nroaming(E) denotes the number of roaming states and ρ(E) is the usual molecular density states, at the total energy E. The expectation is that Nroaming(E) is less than the total number of states of the saddle point, which in this context serves as a convenient reference configuration for the wavefunction calculations.

3 Summary and conclusions

In this review dynamical and non-dynamical approaches to roaming reactions have been presented with a focus on the seminal H2CO reaction. Recent quasi-classical trajectory calculations were presented, using a new global potential energy surface. Although it is likely that similar QCT calculations will continue to provide key information about and validation of roaming, it is also clear that they will provide neither a complete picture of the phenomenon nor a simple method for calculating rate constants. However, one can be encouraged by the variety of other theoretical approaches being pursued, as reviewed above.

A deeper quantum mechanical understanding of roaming is an obvious need. Wave packet propagation will be certainly be important for small systems, and for larger ones it is likely that spectroscopic approaches can help, but in either case the techniques need to be developed for wider use.

For some time it has been clear that a phase space analysis can provide important information that is unavailable from consideration of the potential energy features of configuration space without as well considering and analyzing the associated momenta. Nonetheless, this fact is still under-appreciated by most people studying roaming. Given the previous success of configuration space theories that emphasize the transition state, it is no wonder that there is resistance to largely unfamiliar new analytical concepts. Recall, however, that roaming itself was an unfamiliar concept until 2004. It should not be completely surprising that older theories, in particular those devised for low-level excitation, should now need to be updated with new concepts.

Even acknowledging that phase space analysis may be needed, there is still a strong desire to incorporate roaming into a more traditional kinetic framework, as shown by much of the work reviewed here. Is there an RRKM theory of roaming? The answer to this question is still unclear, but one can be encouraged by the variety of approaches being taken, including “geodesics”, attempts at statistical and kinetic theories, and considerations of the roaming saddle point and the intrinsic reaction coordinate. Perhaps one or more of these approaches may yet provide a simple extension of RRKM theory that will explain how roaming affects rate constants.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank Greg Ezra and Hua Guo for supplying figures used here. We benefited from numerous discussions with colleagues about roaming, especially recent ones with Richard Stratt, Stephen Wiggins, Barry Carpenter and Greg Ezra. We thank our recent co-workers, notably Xiahong Wang and Riccardo Conte for carrying out the recent computational research on roaming in H2CO. The Army Research Office is thanked for partial funding of the recent research on roaming.

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