Theoretical studies on bimolecular reaction dynamics

Abstract

This perspective discusses progress in the theory of bimolecular reaction dynamics in the gas phase. The examples selected show that definitive quantum dynamical computations are providing insights into the detailed mechanisms of chemical reactions.

Afundamental interest in the study of chemical reactions is understanding how molecular quantum states influence reactivity. The simplest examples are reactions in the gas phase between atoms and diatomic molecules

where v and j are vibrational and rotational states, respectively. Crossed molecular beam and laser experiments are enabling these reactive transitions to be measured to an increasing precision as is demonstrated here and in other articles in this issue of PNAS. A major aspect is studying how individual quantum states of the reactant molecules influence the rate of reaction and which product quantum states are produced. Other fundamental details of reactions that can be measured include the dependence on collision energy of the reactant molecules and the angular distributions (differential cross-sections) of the products of the reaction. Sophisticated theory is required to interpret the results of these experiments, and there has been considerable progress in recent years (1, 2). Indeed, theory has progressed so that several detailed predictions have been confirmed by experiment. In turn, new experiments have demanded the need for more detailed and accurate theory (3). One of the exciting features of this field is that surprising results continue to be discovered even on the simplest reactions. This article gives examples of these findings.

Theoretical Approach

The Born–Oppenheimer is invariably invoked, which allows the separation of electronic and nuclear motion. Thus a potential energy surface is first calculated by solving the Schrödinger equation for the electrons with fixed positions of the nuclei. For reliable predictions, these electronic structure computations normally have to be done by using the most accurate ab initio methods that allow for electron correlation so that the crucial transition-state region of the reaction is described accurately (4). Once the potential energy surface is calculated and fitted to a suitable functional form then, in the most accurate quantum dynamics theory, the time-independent or time-dependent Schrödinger equation is solved for the nuclei by using quantum scattering theory. This theory provides all of the fundamental information needed to predict the results for comparison with essentially any experiment on chemical reactions. The reader is referred to a recent review on the theory of reactive scattering that gives >1,000 references and no attempt is made here to reference every aspect of the field (1).

From the point of view of theoretical analysis it is convenient to separate bimolecular reactions into three different types depending on the topology of the potential energy surface going from reactants to products: reactions with a simple barrier (Fig. 1A), reactions with no barrier (Fig. 1B), and reactions with more complicated potentials that can have multiple minima and maxima (Fig. 1C).

Fig. 1.

Potential energy surface profiles plotted against reaction coordinate for reaction with a barrier (A), reaction with no barrier (B), and reactions with more complicated potentials (C).

Different theoretical approaches are needed for these three types of reactions, and illustrative examples of calculations are discussed.

Reactions with a Barrier

Many quantum dynamics calculations have been published for simple reactions, with potential profiles having a simple barrier, between atoms and diatomic molecules (1, 2). For the benchmark chemical reaction

for collision energies at which only one electronic state can be accessed, quantum dynamics theory has been successful in explaining the experimental results including state-selected differential cross-sections for well defined collision energies and more macroscopic quantities such as rate constants (1–3). This comparison applies also to the replacement of H atoms in the reaction by D atoms.

A good example of the exquisite agreement that can now be obtained between theory and experiment is shown in Fig. 2 where differential cross-sections for the

reaction are shown as a function of collision energy for a chosen scattering angle (5). It is seen that these distributions have an oscillatory structure. This structure is associated with the interference between quantum scattering states that have energies close to the effective barriers for different internal states at the transition state (6). These quantum barrier states have been invoked in many studies of chemical kinetics including modern transition-state theory (7) and the classic Rice–Ramsperger–Kassel–Marcus theory of unimolecular reactions (8). Now there is strong evidence they influence the mechanism of the simplest bimolecular reactions.

Fig. 2.

Experimental differential cross-section (dots) for H + D₂ (v = 0, j = 0) → HD (v′ = 0, j′ = 2) + D measured at the laboratory angle of 70°, versus the collision energy, E_C. The curve is the result of the quantum scattering calculation. [Reproduced with permission from ref. 5 (Copyright 2003, American Association for the Advancement of Science).]

Another quantum effect that has been suggested to influence chemical reactions is the geometric phase. This phase can occur in chemical reactions when there is a touching of two electronic states known as a conical intersection. When the nuclei complete an odd number of loops around the conical intersection, the electronic wavefunction must change sign. The geometric phase effect on the H + H₂ reaction, and isotopic variants, has been the subject of several articles (9, 10) and it was suggested that it may be observable in differential cross-sections for scattering energies well below that of the conical intersection (9). However, an elegant topological argument followed up by detailed quantum scattering and classical trajectory calculations has recently explained convincingly that this effect should not be observable for this reaction at these energies (11).

The reaction

has been of special interest to theoreticians for many years because of the strong effect of Feshbach resonances that are predicted for this reaction by using even very approximate quantum scattering treatments (12). These resonances occur when the shape of the potential surface and internal-mode couplings cause reactant flux to be trapped temporarily in a region close to the transition state of the reaction. In turn, these resonances are predicted to produce sharp peaks at certain angles in the differential cross-section when plotted as a function of collision energy (13). Several molecular beam experiments on these reactions have suggested the possibility of observing these resonances (14) and, as shown in Fig. 3, very recent and highly resolved experiments now provide clear evidence for them (15). In both theory and experiment, there is a sharp rise in the differential cross-section over a narrow range of collision energies and scattering angles associated with a sudden excitation into product HF (v = 3). Indeed the agreement between theory and experiment for all details of the differential cross-section is exceptional. It should be emphasized that considerable work is required in refining the quantum chemistry calculations needed for an accurate description of the potential energy surface for this reaction and spin-orbit coupling between the ²P_3/2 and ²P_1/2 electronic states of the F atom needs to be treated in the most sophisticated treatments (16).

Fig. 3.

Experimental (A) and theoretical (B) 3D contour plots for the product translational energy and angle distributions for the F + H₂ → HF + H reaction at the collision energy of 0.52 kcal/mol (15). The different circles represent different HF product ro-vibrational states. The forward-scattering direction for HF is defined along the F-atom beam direction. [Reproduced with permission from ref. 15 (Copyright 2006, American Association for the Advancement of Science).]

Although the reactions of H₂ with H and F atoms have been the subject of the most detailed theoretical and experimental investigations, other reactions between atoms and diatomic molecule with simple barriers are also beginning to receive similar attention (2). This includes

where there is a continuing debate between experiment and theory on the importance of nonadiabatic transitions between the electronic states on this reaction (17).

The reaction,

has also been the subject of detailed experiment and quantum dynamics theory (18, 19). The particular interest here is determining which particular vibrational states of the H₂O triatomic product are produced in the reaction (18). Predictions of these vibrational states for the

reaction using a quantum scattering theory with a potential energy surface fitted with ab initio data gave excitation predominantly in the doubly excited OD stretching vibration of HOD (20). As shown in Fig. 4, these predictions were closely confirmed in subsequent experiments (21). Other calculations have treated this reaction in full dimensionality and have allowed comparisons with a variety of experiments including differential cross-sections, integral cross-sections, and rate constants (22).

Fig. 4.

Predicted (20) and measured (21) vibrational product distributions for the OH + D₂ → DOH + D reaction at a collision energy of 6.3 kcal/mol. The integer labels n, m (e.g., 0, 0) are the DOH bending (n) and OD stretching (m) vibrations of product DOH.

The success of the theoretical predictions on the reactions of H₂ with H, F, and OH has stimulated several calculations on the dynamics of more complicated reactions with polyatomic molecules as reactants or products (see refs. 1 and 2 for reviews). These systems present two major difficulties. First, constructing an accurate potential energy surface from ab initio electronic structure calculations for reactions with more than four atoms is not straightforward. As a result some novel and highly efficient fitting techniques have been developed to minimize the number of ab initio points that need to be calculated (23, 24). Second, treating all of the degrees of freedom of the reactant or product molecules makes the dynamics calculations expensive to carry out computationally; there are 3N − 6 vibrational modes for an N-atom system. New dynamical approaches are being developed such as couple coherent states (25) and multiconfigurational time-dependent Hartree (MCTDH) (26) that are enabling quantum dynamical calculations to be done for many coupled degrees of freedom. As an example, MCTDH calculations of rate constants for the

reaction using an ab initio potential surface gave very good agreement with experiment (26). The ideal quantum dynamics theory would avoid fitting the potential energy surface in the first place and would just call the ab initio computer code whenever such data are needed. However, as quantum wavefunctions cover a very wide region of configuration space such calculations would be very expensive, especially for polyatomic reactions. With this in mind, a method has been developed that allows for the construction of efficient reduced dimensionality potential surfaces with a minimum number of accurate ab initio points (typically ≈50) followed by time-independent reduced-dimensionality quantum scattering calculations for the reaction dynamics treating the chemical bonds being broken and formed in the reaction explicitly (27). This procedure has been carried out for a variety of polyatomic reactions that include different chemical products and for which analytical potentials are not available. For example, Fig. 5 shows a comparison of calculated (27) and observed branching ratios for producing the two chemically distinct iso- and n-propyl radicals in the reaction

Many calculations on reactions such as this have shown that quantum tunneling is important for reactions with barriers involving H atom transfer and quantum dynamics theories with the most accurate potential surfaces need to be used to predict reliable rate constants. Properties of chemical reactions such as product state distributions can be less sensitive to details of the potential energy surface and more approximate dynamics theories and electronic structure theories can then be used quite effectively as is discussed below.

Fig. 5.

Variation with temperature of the different contributions from the two product channels to the overall reaction H + C₃H₈ → H₂ + CH₃CHCH₃ (iso), CH₂CH₂CH₃ (n). Present refers to the calculations and the other curves to recommended experimental results (see ref. 27).

Reactions with No Barrier

Reactions between ions and molecules or those between open-shell atomic and molecular radicals often have no barriers in the potential energy surface between reactants and products. These fast reactions have rate constants typically in the range 10⁻⁹ to 10⁻¹¹ cm³·s⁻¹·mol⁻¹ that are several orders of magnitude larger than those for the reactions with barriers discussed above. Fits to the temperature dependences of rate constants for barrierless reactions do not have the standard Arrhenius form and sometimes they decrease with increasing temperature (28).

For these reactions, which are frequently very exothermic, a capture theory can often be applied that allows for consideration of only the part of the entrance channel of the potential energy surface that contains the centrifugal potential associated with the rotation of the reactant complex (29, 30). Applying the capture assumption, that trajectories with energy above the effective barrier will not recross the transition state, allows for simple procedures to calculate reaction rate constants and their initial quantum state dependences. When the potential energy surfaces are known in this long-range region, such as for reactions of ions with molecules, analytical rate constants can be derived that have simple dependences on the properties of the reactant molecules. The best-known such example is the Langevin cross-section for the reactions of ions with nonpolar molecules that gives a rate constant with no temperature dependence (29). The open-shell nature of the reactant species can also be treated readily by using capture approximations, which can lead to different analytical predictions than for closed-shell species of the rate constants in the low-temperature limit (30–32). Furthermore, other theories can be combined with the capture assumption. For example, when there is more than one reaction product a combined capture-wavepacket method can be applied to predict product branching ratios (33). The capture ideas have also been combined with time-independent quantum scattering theory to predict the ro-vibrational product distributions of barrierless chemical reactions such as C(¹D)+ H₂ (34).

The field of astrophysical chemistry has benefited much from the theoretical predictions on these fast reactions. In diffuse interstellar clouds the temperatures can be as low as 10 K and only reactions without barriers in the potential can then be significant in the reaction networks that describe how molecules are produced. The role of ion–molecule reactions in this area was known for some time but it was only following more recent theoretical and experimental studies that reactions of neutral species were also shown to have large rate constants at these low temperatures and to be important in these networks (35). The reactions of C(³P) reactions with alkenes and alkynes are good examples as these enable longer chain carbon molecules to be produced at low temperatures, and the observed large rate constants are readily explained by using capture theory (36).

A very recent combined theoretical and experimental study of the reaction between O(³P) and alkenes illustrates systematically the transition from potentials with quite a large barrier to those with almost no barrier as the size of the alkene is increased (37). Thus the reaction of O(³P) with ethene is slow, whereas those for propene and especially butene are relatively fast, and the rate constants do not increase in a simple form as the temperature is increased (see Fig. 6). A microcanonical transition-state theory that treats two types of transition states together with ab initio calculations of key points on the longer-range region of potential energy surface explains the experimentally observed trends well (37).

Fig. 6.

The points show experimentally determined values of the rate constants for the reactions of O(³P) atoms with alkenes at different temperatures, and the dashed lines show the results of calculations. The solid lines represent the recommended Arrhenius expressions. [Reproduced with permission from ref. 37 (Copyright 2007, American Association for the Advancement of Science).]

Reactions with Complicated Potentials

Reactions with several maxima and minima in the potential energy surface present the most severe challenge to calculating and fitting potential energy surfaces and carrying out quantum dynamics calculations. Even atom + diatom reactions such as

can have this kind of complicated potential surface, and this reaction has been the subject of several quantum-dynamical calculations for comparison with detailed molecular beam experiments (as reviewed in refs. 1–3). One channel of this highly exothermic reaction correlates with the bound H₂O molecule, and the potential surface includes a deep well. The picture emerging for the mechanism of the reaction [and that of O(¹D) + HD] involves an insertion into the middle of the H₂ bond via a long-lived complex at low collision energies, with the characteristic forward and backward shape of the angular distribution, whereas at higher energies the mechanism is closer to abstraction of an H atom with more pronounced backward scattering of the differential cross-section (3, 38).

A small number of approximate quantum scattering calculations have been performed on reactions involving larger molecules with complicated potentials. For example reduced dimensional calculations on the classic S_N2 reaction

suggest that reaction occurs via many scattering resonances correlating to excited ion–dipole complexes (39, 40). However, in probing the dynamics of these and more complicated reactions the use of quantum dynamics methods is limited because of the number of states that need to be included in the computations and it is often necessary to resort to methods based on classical dynamics. This approach of molecular dynamics is now used quite routinely to simulate chemical processes throughout chemistry, biochemistry, and materials science. It can be applied to systems involving thousands of atoms, including those in condensed phases (41). Many of these calculations use approximate molecular mechanics force fields, and if chemical reactions are of interest the results have to be treated with care because of the difficulties of describing reaction potential energy surfaces accurately (4). Density functional theory (DFT), although normally giving only an approximate description of reaction potential surfaces, can be combined very efficiently with classical mechanics. This method and other approximate electronic structure methods have been implemented in the method called direct dynamics that avoids the difficulty of constructing an analytical form for the potential energy surface and calculates energies and potential derivatives needed for solving the classical trajectory equations “on the fly” (42). The Car–Parrinello method is the most widely used direct dynamics method approach that combines DFT directly with molecular dynamics (43).

Results of such classical dynamics calculations need to be considered with special care for reactions involving hydrogen atom transfer where quantum tunneling can be important. New semiclassical approaches have promise for these cases (44). If just rate constants are needed, then sophisticated forms of variational transition-state theory with tunneling corrections can also be applied very effectively. These methods have been tested and checked for accuracy on the benchmark atom–diatom reactions (45) and are now being applied to reactions of biochemical importance. A good example is application to the coupling of hydrogenic tunneling to active-site motion in the hydrogen radical transfer catalyzed by a coenzyme (46). Recently, a hybrid quantum–classical path integral method has also been applied effectively to this problem (47).

Conclusion

The ideal theory for treating bimolecular reactions involves accurate quantum dynamics using accurate potential energy surfaces. As has been described here, calculations with this approach can be done on reactions involving up to four atoms but more approximate methods are normally needed for more complicated reactive systems. These definitive calculations are providing insight into the detailed mechanisms of chemical reactions.

As methods are developed and computers get more efficient it is inevitable that the range of the rigorous quantum theory will improve. A key aspect will be the interface with the most accurate electronic structure methods to produce truly direct quantum dynamics methods that can be applied to more complicated reactions for which analytical potential energy surfaces are not available. Another area where there is a need for intensive research is in the rigorous treatment of transitions between electronic states in chemical reactions. Even for simple benchmark reactions such as F + H₂ this treatment is needed to explain some experimental results. There is no doubt that the theoretical study of reaction dynamics will be a very active research area for many years to come.