Perspective: chemical dynamics simulations of non-statistical reaction dynamics

Abstract

Non-statistical chemical dynamics are exemplified by disagreements with the transition state (TS), RRKM and phase space theories of chemical kinetics and dynamics. The intrinsic reaction coordinate (IRC) is often used for the former two theories, and non-statistical dynamics arising from non-IRC dynamics are often important. In this perspective, non-statistical dynamics are discussed for chemical reactions, with results primarily obtained from chemical dynamics simulations and to a lesser extent from experiment. The non-statistical dynamical properties discussed are: post-TS dynamics, including potential energy surface bifurcations, product energy partitioning in unimolecular dissociation and avoiding exit-channel potential energy minima; non-RRKM unimolecular decomposition; non-IRC dynamics; direct mechanisms for bimolecular reactions with pre- and/or post-reaction potential energy minima; non-TS theory barrier recrossings; and roaming dynamics.

This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.

1. Introduction

Statistical theories in chemical dynamics and kinetics are theoretical models whose assumptions lead to the use of statistical mechanics to calculate properties for chemical reactions, such as rate constants and product energy partitioning distributions [1]. Transition state theory (TST) is derived using classical mechanics and is based on two fundamental assumptions: (i) the reactants have Boltzmann energy distributions and (ii) once a classical trajectory crosses the transition state (TS) towards products, it does not turn around and recross the TS forming reactants (i.e. the no-recrossing assumption of TST). TST is a canonical classical statistical mechanical model for chemical kinetics. Attempts to derive a quantum TST have been unsuccessful [2,3], and Miller has stated ‘there is no rigorous quantum version of transition-state theory’ [2]. The widely used model quantum TST results by replacing the classical partition functions for the TS and reactant(s) with their quantum counterparts.

Rice–Ramsperger–Kassel–Marcus (RRKM) unimolecular rate theory [4–7] is based on the fundamentals of microcanonical statistical mechanics; in the mass spectrometry community, RRKM theory is often referred to as quasi-equilibrium theory [6]. The fundamental assumption of the theory is that a microcanonical ensemble of vibrational states is maintained for the unimolecular reactant as it decomposes to products. RRKM is a TST and the RRKM rate constant is derived using classical mechanics. It also has the no-recrossing of the TS assumption. What is known as quantum RRKM theory results by replacing the sum of states at the TS and the density of states of the unimolecular reactant with their quantum counterparts. RRKM theory is a microcanonical statistical mechanical model for unimolecular kinetics.

An important component of statistical models of chemical reactions is the assumed motion on the potential energy surface (PES) before products are formed [8]. The nature of the dynamics on the PES, before reaching a rate-controlling TS, does not affect the reaction kinetics, but the dynamics after passing this TS are critical. They are referred to as ‘post-TS’ dynamics [9,10]. To predict attributes of chemical reactions after passing this TS, non-TST models are needed. Post-TS dynamics are determined by features of the PES, and they include branching between different product channels and reaction pathways and partitioning of the available energy to the reaction products.

The PES in this post-transition state domain may have a very ‘rough’ landscape, with multiple potential energy minima, reaction pathways, low energy barriers, etc., connecting the rate-controlling TS to multiple product channels. The important question is ‘what are the post-TS dynamics?’ A statistical model is often used to model the post-TS dynamics. For this model, potential energy minima, barriers and different product channels are connected via reaction paths determined by following their intrinsic reaction coordinates [11]. For the gas phase, where the energy of the reactive system is conserved, the statistical model uses RRKM theory to determine the time the system spends in each potential energy minima and the probabilities for transitions between minima and forming products. The statistical model for gas-phase product energy partitioning is phase space theory (PST) [12–15] or a modified PST [16,17].

An intrinsic reaction coordinate (IRC) is the path of steepest descent from a TS to a potential energy minimum [11]. This motion results from setting the velocity of each atom to zero after each infinitesimal step. As energy is constant for gas-phase events, there is not a gas-phase trajectory that has the IRC motion. The IRC provides well-defined structures between stationary points (i.e. TSs, potential energy minima, reactants and products) on the PES.

Non-statistical dynamics are properties of chemical reactions which lead to violations of the fundamental assumptions of statistical theories, and/or the assumed statistical dynamics for motion on the PES. The following are important illustrative examples of non-statistical reaction dynamics, with chemical dynamics simulations by the author's research group emphasized. Extensive non-statistical dynamics have been observed, both experimentally and computationally, for X⁻ + CH₃Y → XCH₃ + Y⁻ S_N2 bimolecular substitution reactions [18] and a number of the examples discussed below are for these reactions. As illustrated in figure 1, the PES for these reactions have X⁻‐‐‐CH₃Y pre- and XCH₃‐‐‐Y⁻ post-reaction complexes and a [X‐‐CH₃‐‐Y]⁻ central barrier TS.

Figure 1.

Potential energy curve for the Cl⁻ + CH₃I → ClCH₃ + I⁻ reaction. The energies are given by MP2/ECP/aug-cc-pVDZ theory. Adapted from [8,18].

2. Illustrative examples of non-statistical dynamics

(a). Post-transition state dynamics

If a rate-controlling TS provides an accurate rate constant for the chemical reaction (i.e. TS recrossing is unimportant), an ensemble of trajectories may be initiated at the TS and propagated to products to study the post-TS dynamics [10]. For constant temperature unimolecular or bimolecular reactions, the initial conditions for these trajectories are chosen in accord with quantum TST. For a constant energy unimolecular reaction, for which the unimolecular dynamics are statistical and in accord with RRKM theory, to study the post-TS dynamics initial conditions are chosen at the TS as specified by quantum RRKM theory [19]. Quasi-classical initial conditions are chosen to sample the TS's quantum mechanical energy levels and explicitly include the TS zero-point energy (ZPE). The following are examples of non-statistical post-TS dynamics.

(i). Bifurcations and branching between exit-channel pathways

For a reaction whose PES has a bifurcation after passing the rate-controlling TS, chemical dynamics simulations are critical for predicting the non-statistical product branching ratio [20–25]. The IRC [11] leads to a valley-ridge inflection (VRI) point and then dynamics are needed to determine the branching between the two product channels. A model PES has been developed to study non-statistical dynamics associated with PESs with reaction path bifurcations and VRI points [24]. Simulations on this PES [24] indicate that, in the gas phase, dynamics in the vicinity of the VRI are insignificant in determining the product branching; however for a reactive system with a strongly interacting solvent, and extensive energy transfer from the reactive system, the VRI may play an important role. Potential energy release to the solvent, in moving from the TS to the VRI, will tend to constrain the motion near the vicinity of the IRC and, thus, make dynamics in the vicinity of the VRI more important.

(ii). Product energy partitioning

Repulsive potential energy release. If there is repulsive potential energy release as the reactive system moves directly from the rate-controlling TS to products, statistical theory cannot be used to accurately model the product energy partitioning. Product energy partitioning for such a case is illustrated by the direct dynamics simulations for C₂H₅F → HF + C₂H₄ dissociation [19,26,27]. The simulations were performed using MP2 theory with both the 6-31G* and 6-311++G** basis sets, and the results are compared with experiments of Setser, Wittig and co-workers [28,29]. The experimental 0 K potential energy release with ZPE included, E_r^o, in going from the TS to products is 49.0 ± 2.0 kcal mol⁻¹, while the MP2 values with the 6-31G* and 6-311++G** basis sets are 45.3 and 50.6 kcal mol⁻¹, respectively.

An important comparison with experiment is the partitioning of the available product energy to HF vibration. For the experiments, excess energies at the TS (E_vt^‡) of 32 and 42 kcal mol⁻¹ were investigated. For the latter E_vt^‡, the experiments show that 15% of the available energy is partitioned to HF vibration. The MP2/6-311++G** simulation gives 14% and nearly the same result, while the MP2/6-31G* simulation gives 8%. A more detailed comparison with experiment is the population distribution, P(n), of the HF vibrational levels, which is shown in figure 2. The MP2/6-311++G** simulations are in very good agreement with both the E_vt^‡ of 32 and 42 kcal mol⁻¹ experiments. Their principal difference is the small P(1)/P(0) inversion found from the simulations.

Figure 2.

Populations of the HF vibrational states for C₂H₅F → HF + C₂H₄ dissociation, with different amounts of vibration/reaction coordinate energy E_vt^‡ in the C₂H₅F^‡ transition state. Open squares, results of the MP2/6-311++G** simulations; filled circles, results of the MP2/6-31G* simulations from [17]; filled triangles, experimental results. The experimental results for E_vt^‡ of 32 and 42 kcal mol⁻¹ are from [21] and [20], respectively. Adapted from [26].

Also of interest are the percentages for partitioning to the products relative translation, rotation and vibration energies, obtained from the simulations but not the experiments. The results are given in table 1. The majority of the product degrees of freedom are for C₂H₄ vibration and for a statistical model the majority of the product energy would be partitioned to C₂H₄ vibration. By contrast, as shown in table 1, the majority of the energy is distributed to HF + C₂H₄ relative translation, arising from non-statistical, repulsive potential energy release in moving from the TS to products. In comparing the results of the two basis sets, 6-311++G** partitions slightly less to relative translation and more to HF vibration, with the other energy partitionings similar for the two basis sets.

Table 1.

Average product energy partitioning for C₂H₅F^‡ → HF + C₂H₄ direct dynamics^a.

		MP2/6-31G*			MP2/6-311++G**
product energy	Evt‡=	3.45	27	42	3.45	32	42
rel trans		75.4	56.8	49.7	67.8	48.2	46.2
C2H4 vib		6.1	24.2	31.5	6.8	26.2	29.3
C2H4 rot		4.6	5.9	6.0	5.1	5.5	6.2
HF vib		10.5	8.7	7.9	16.9	15.8	14.0
HF rot		3.4	4.4	4.9	3.4	4.3	4.3

^aThe average energy partitioning is given in per cent. E_vt^‡ is in kcal mol⁻¹.

Following a model proposed by Zamir & Levine [30], the product energy partitioning was analysed versus E_vt^‡ to determine the percentages of the potential energy release, E_r^o, distributed to the different product energies. For the MP2/6-31G* simulations, the percentages partitioned to relative translation, C₂H₄ vibration, C₂H₄ rotation, HF vibration and HF rotation are 81, 0, 5, 11 and 3%, respectively. For the MP2/6-311++G** simulations, these percentages are 73, 2, 4, 18 and 3%. Thus, the vast majority of E_r^o is partitioned to relative translation, with HF vibration coming in a distant second.

Calculating a single trajectory (ST) initiated at the TS, with a very small reaction coordinate translational energy and no energy in the TS's vibration (including ZPE) and rotation degrees of freedom, gives semi-quantitative values for partitioning of E_r^o to product energies. This model was used to study the effect of mass on energy partitioning for ‘C₂H₅F → HF + C₂H₄’ dissociation, where the MP2/6-31G* PES for C₂H₅F was used but masses of some of the atoms were changed for the ST direct dynamics simulations [27]. For the ST, 1.0 kcal mol⁻¹ was added to reaction coordinate translation and the results of the simulations are given in table 2. Only changing the mass of F to Cl has a negligible effect on the product energy partitioning. However, changing the masses of the substituent H atoms to the mass of Cl has a dramatic effect. For the CHCl₂CH₂Cl model, where −CH₂Cl contains the dissociating Cl atom, partitioning of energy to relative translation is dramatically lowered to 44%, with 14% and 26% partitioned to CCl₂CH₂ vibration and rotation, respectively. Changing the model to CH₃CCl₃ from CHCl₂CH₂Cl decreases and increases, respectively, energy partitioning to the olefin's rotation and vibration, with only a small change in the partitioning to relative translation. For CHCl₂CCl₃, with all of the substituent atoms Cl except one, the energy releases to relative translation and C₂Cl₄ vibration are nearly equivalent, while the release to C₂Cl₄ rotation is essentially zero.

Table 2.

Product energy partitioning for single trajectories using the MP2/6-31G* C₂H₅F → C₂H₄ + HF PES^a.

molecule	rel trans	vibb	rotb	HX vib	HX rot
C2H5F	74.9	6.8	1.5	14.4	2.4
C2H5Cl	75.6	5.9	2.4	15.9	0.2
CHCl2CH2Cl	43.9	14.2	26.0	15.8	0.1
CH3CCl3	47.3	24.4	11.5	15.7	1.1
CHCl2CCl3	39.7	38.1	0.2	16.1	5.9

^aThe single trajectories were initiated at the saddlepoint with only 1.0 kcal mol⁻¹ in reaction coordinate translation, E_t^‡.

^bVib and rot are the vibration and rotation energies of the ethylene product, respectively.

The position of the centre of mass of the dissociating molecule affects partitioning of energy to olefin rotation for CHCl₂CH₂Cl, CH₃Cl₃ and CHCl₂CCl₃. Of these three molecules the largest percentage partitioning to olefin rotation is for CHCl₂CH₂Cl. For this molecule, the dissociation occurs by the H-atom of CHCl₂ forming HCl with the Cl-atom of CH₂Cl and then the HCl product recoils from the C-atom of the CH₂ group of the product olefin. The centre of mass of the product olefin is located near the Cl atoms of the CCl₂ group and the above recoil results in substantial rotation about this centre of mass, which is distant from the recoil site. For CH₃CCl₃, the product olefin's centre of mass is near the Cl atoms of the CCl₂ group, but also near the recoil site (i.e. the C-atom of this group) and as a result energy transfer to olefin rotation is much smaller. The CCl₂CCl₂ olefin product is symmetric, for CHCl₂CCl₃ dissociation, with heavy atoms at both of its ends and a very large moment of inertia. It receives negligible rotational energy.

There are also significant mass effects for vibrational excitation of the olefin product. For C₂H₅F dissociation, only 6.8% of the product energy goes to olefin (i.e. C₂H₄) vibration, while for the CCl₂CCl₂ olefin product it is 38.1%. Analyses of the trajectories for CHCl₂CCl₃ → HCl + CCl₂CCl₂ dissociation show that two-thirds to three-fourths of the vibrational energy in the CCl₂CCl₂ product resides in the out-of-plane CCl₂ wag-bend motions. There are two significant factors for this large wag-bend excitation and both are related to the mass of the Cl-atoms. As HCl dissociates from CHCl₂CCl₃, there is very little change in the CCl₂ wag angles from their sp³ values for the dissociating molecule. HCl dissociates rapidly and there is insufficient time for the heavy Cl-atoms to move appreciably. A Franck–Condon type model [31–33] approximates the resulting deposition of vibrational energy in the CCl₂ wag-bends.

The other factor arises from the repulsive interaction between the Cl-atom of HCl and the C-atom from which Cl dissociates. If the C-atom is attached to two Cl-atoms, as HCl pushes off the C-atom, the Cl-atoms bonded to it move very little because of their substantial mass and the CCl₂ wag-bend receives kinetic energy from the C-atom's motion. Another rather small Franck–Condon factor is also a mass effect. The Cl-atoms restrict the motion of the C-atoms, resulting in some vibrational excitation in the C=C stretch.

Of the three chlorinated model olefins, the energy partitioning to olefin vibration is 14.2, 24.4 and 38.1% for CHCl₂CH₂Cl, CH₃Cl₃ and CHCl₂CCl₃, respectively. These percentages are consistent with the factors discussed above. For CHCl₂CH₂Cl dissociation, the important factor for vibrational excitation of the CCl₂CH₂ product is the Franck–Condon excitation of the CCl₂ wag-bend. For CH₃Cl₃ dissociation, there is this wag-bend excitation as well as the repulsive interaction between the dissociating Cl-atom and the C-atom of the CCl₂ group. For CHCl₂CCl₃ dissociation, there are three factors: this repulsive interaction and Franck–Condon excitations of the two CCl₂ wag-bends.

Exit-channel potential energy minimum. Many PESs have a potential energy minimum (or minima) in the exit channel connecting the TS and products. The statistical model for the exit-channel dynamics assumes the trajectories become trapped in these minima forming intermediates which, after rapid and complete intramolecular vibrational energy redistribution (IVR), dissociate to products with RRKM kinetics. If there is no reverse barrier for association of the reaction products to form the intermediate, the statistical model assumes there is a statistical distribution of energy for the reaction products as specified by PST [12–15] or a modified PST [16,17]. However, this model may be inaccurate and the assumption of RRKM dynamics for the intermediate is then not correct.

An illustration of this type of non-statistical dynamics is found for the Cl⁻ + CH₃Br → ClCH₃ + Br⁻ S_N2 reaction, which has a ClCH₃‐‐‐Br⁻ complex in the exit channel [10,34]. If this complex is formed and its dynamics statistical, the ClCH₃ + Br⁻ product energy partitioning should agree with the prediction of PST. In an experimental study of dissociation of the Cl⁻‐‐‐CH₃Br pre-reaction complex, Graul & Bowers [35,36] measured the relative translational energies between the ClCH₃ + Br⁻ reaction products and found them to be much smaller than the prediction of PST. A chemical dynamics simulation was performed to model these experiments by exciting the Cl⁻‐‐‐CH₃Br complex and studying its dissociation. The intramolecular and unimolecular dynamics of this complex, and of the ClCH₃‐‐‐Br⁻ exit-channel complex, were studied and found to be non-RRKM [10]. The post-TS dynamics after crossing the [Cl‐‐CH₃‐‐Br]⁻ central barrier is decidedly non-RRKM. The average product relative translational energy found from the simulations is 0.8 kcal mol⁻¹, in excellent agreement with the experimental value of 0.7 ± 0.2 kcal mol⁻¹ and considerably different than the PST value of 1.6 kcal mol⁻¹.

(iii). Avoiding exit-channel potential energy minima

Though there may be deep potential energy minima in the exit channel for chemical reactions, in the multi-dimensional post-TS dynamics the trajectories may avoid, i.e. not find, these minima. These dynamics are found for the F⁻ + CH₃OH and F⁻ + CH₃OOH reactions [37,38]. Energies and geometries for the potential energy curve of the F^- + CH₃OH S_N2 reaction are shown in figure 3. The statistical model assumes the post-TS dynamics are indirect with the system temporarily trapped in the CH₃OH‐‐‐F⁻ minimum. To test this model, direct dynamics trajectories were initiated at the central barrier TS, with conditions chosen from a 300 K Boltzmann distribution. Two reaction pathways were found for the trajectories, one direct and the other indirect. The vast majority, approximately 90%, follow a direct reaction pathway with departure of the F^- ion approximately along the O-C‐‐‐F⁻ collinear axis, avoiding the potential energy minimum for the CH₃OH‐‐‐F⁻ intermediate. The remaining small fraction of trajectories initially follow the direct pathway, but they do not have sufficient CH₃OH + F⁻ relative translational energy to dissociate and are drawn into the CH₃OH‐‐‐F⁻ minimum. As the system moves off the central barrier, it is propelled towards products by the potential energy release and does not move towards the potential energy well to form the CH₃OH‐‐‐F⁻ intermediate.

Figure 3.

Potential energy along the IRC for OH⁻ + CH₃F → CH₃OH + F⁻. s is the distance along the IRC. The potential energy and IRC were calculated at the MP2/6-311G* level of theory. Adapted from [37].

Similar dynamics in which a deep exit-channel minimum is avoided is found for the F⁻ + CH₃OOH reaction [38]. In experimental studies of this reaction [39], it was found that the most exothermic products HF + CH₂(OH)O⁻, and products predicted by the potential energy curve for the reaction (figure 4), are not formed. Instead, their experiments are consistent with formation of the much higher energy products HF + CH₂O + OH⁻ by an E_CO2 mechanism. To provide an atomic-level understanding of the experimental results, a direct dynamics simulation was performed [38]. Trajectories were initiated for the F⁻ + CH₃OOH reactants with initial conditions to model the 300 K experiments. The post-TS dynamics for the trajectories, after passing TS1, did not lead to the deep potential energy of the CH₂(OH)₂‐‐‐F⁻ intermediate and instead directly formed the HF + CH₂O + OH⁻ products by the proposed E_CO2 mechanism. From the trajectories, the branching between the HF + CH₂O + OH⁻ and HF + CH₃OO⁻ product channels is predicted to be 0.98 ± 0.01 and 0.02 ± 0.01, fractions in qualitative agreement with the experimental estimates of approximately 85 and 10% for these two channels. The trajectory estimate of only approximately 2% branching to the HF + CH₃OO⁻ products is not inconsistent with the experiments, based on a difficult numerical analysis. The trajectory total reaction rate constant for F⁻ + CH₃OOH is (1.70 ± 0.7) × 10⁻⁹ cm³ molecule-s⁻¹ and in excellent agreement with the experimental value of 1.23 × 10⁻⁹ cm³ molecule-s⁻¹. As an additional test of the post-TS dynamics, trajectories initiated at TS1 with the energy of the reactants go directly to HF + CH₂O + OH⁻ and to not move to the deep potential energy minimum.

Figure 4.

Energy diagram for the F⁻ + CH₃OOH reaction at the B3LYP/6-311+G(d,p) level of theory. The energies shown are in kcal mol⁻¹ and are relative to the F⁻ + CH₃OOH reactant channel. Zero-point energies are not included. The percentages are for the trajectories remaining in these regions of the PES when the B3LYP/6-311+G** direct dynamics trajectories are concluded at 4 ps. Adapted from [38]. (Online version in colour.)

The non-statistical post-TS dynamics found for the F⁻ + CH₃OOH reaction is illustrated by the two-dimensional potential energy diagram in figure 5 [38]. Q₁ represents the concerted movement of HF and OH⁻ away from CH₂O, and Q₂ represents the in-plane rotation motion of CH₂O to access the deep potential energy minimum. Shown by the black line in figure 5 is a representative trajectory, which ‘skirts’ the deep potential energy minimum of the CH₂(OH)₂‐‐‐F⁻. From the dynamics, the concerted movement of HF and OH⁻ away from CH₂O is much faster than the rotation motion of CH₂O needed to access the potential energy minimum. Also, there is no barrier restricting the movement of HF and OH⁻ away from CH₂O. The red line illustrates the IRC leading to the complex. The dynamics for the F⁻ + CH₃OOH reaction are reminiscent of those above for the OH⁻ + CH₃F → CH₃OH + F⁻ reaction.

Figure 5.

A two-dimensional contour diagram of the post-transition state PES for TS1. Q₁ = Δr₁ + Δr₂, where r₁ is the FH–C bond length and r₂ is the O─OH bond length. Q₂ = Δθ₁ + Δθ₂, where Δθ₁ is the O–C–O angle and Δθ₂ is the H–O–C angle; i.e. H is the hydrogen abstracted by F⁻, and O is the oxygen attached to carbon. Q₁ represents the concerted motion of HF and OH⁻ away from CH₂O, and Q₂ represents rotation of CH₂O. The remaining coordinates were optimized for each Q₁, Q₂ point. Depicted on this contour diagram is the IRC (red line) and a representative trajectory (black line). Adapted from ref. [38].

As found from direct dynamics simulation, deep exit-channel potential energy minima are also avoided for the microsolvated ions OH⁻(H₂O) and F⁻(H₂O) reacting with CH₃I [40,41]. The reaction dynamics are similar for these two ions and those for OH⁻(H₂O) are discussed here. The trajectories were initiated for the OH⁻(H₂O) + CH₃I reactants so that all possible reaction pathways could be studied. A schematic potential energy profile for the reaction is shown in figure 6. The reaction has a potential energy minimum for the CH₃OH‐‐‐I⁻(H₂O) post-reaction complex for the S_N2 reaction. The statistical model assumes this complex is formed and then dissociates to the various S_N2 products. This complex was not formed in any of the trajectories. The dominant dynamics were for H₂O to dissociate from the system as OH⁻(H₂O) attached to CH₃I, with the associated potential energy release, or dissociate as OH⁻ displaced I⁻ with the much greater potential energy release. For the S_N2 product ions I⁻ and I⁻(H₂O), there is a much greater yield of I⁻. For OH⁻ + CH₃I collision energies in the range of 2.0–0.05 eV, the I⁻(H₂O)/I⁻ ratio varies from 0.011 to 0.021 from the experiments and 0.018–0.041 from the simulations. The post-TS dynamics for the OH⁻(H₂O) + CH₃I reaction are not consistent with a statistical model.

Figure 6.

Schematic energy profile for the OH⁻(H₂O) + CH₃I → CH₃OH + I⁻ + H₂O S_N2 reaction, and other pathways, at the DFT/B97-1/ECP/d level of theory. The energies shown are in kcal mol⁻¹ and are relative to the OH⁻(H₂O) + CH₃I reactants. Zero-point energies are included. Adapted from ref. [40].

(b). Unimolecular decomposition

The statistical model for unimolecular dissociation assumes that IVR is rapid and complete for a unimolecular reactant so that its kinetics is accurately described by RRKM theory [7]. For numerous organic reactions, there is non-statistical unimolecular dynamics [42]. Non-RRKM behaviour is found in both experimental and simulation studies of X⁻‐‐‐CH₃Y pre-reaction complexes for X⁻ + CH₃Y → XCH₃ + Y⁻ S_N2 reactions [18,43]. The energy released, when the reactants X⁻ and CH₃Y associate, is initially only in the three intermolecular modes of the complex. Energy is then transferred to the nine CH₃Y intramolecular modes by slow IVR. For a trajectory simulation of Cl⁻ + CH₃Cl association at 300 K, the number of surviving Cl⁻‐‐‐CH₃Cl complexes versus time, N(t)/N(0), is accurately fit by a sum of three exponentials with rate constants varying from 1.1 to 0.093 ps⁻¹ [44]. RRKM theory predicts a single exponential. When the trajectory N(t)/N(0) is averaged over collisions [45], a lifetime for the complex is obtained which is in excellent agreement with experiment as shown in figure 7 [46,47]. The lifetime does not agree with RRKM theory [45,46].

Figure 7.

Average lifetime of the Cl⁻‐‐‐CH₃Cl complex, extracted from the Cl⁻ + CH₃Cl → Cl⁻‐‐‐CH₃Cl ternary rate constant [46]. Included are values determined by high-pressure mass spectrometry [47] and chemical dynamics simulations [45]. Adapted from [18,46].

The unimolecular dynamics of the I⁻‐‐‐CH₃I complex, formed by I⁻ + CH₃I association, has been studied by time-resolved photoelectron spectroscopy [48]. The time dependence of the I⁻‐‐‐CH₃I complex's lifetime, N(t)/N(0), is multi-exponential as discussed above for Cl⁻‐‐‐CH₃Cl. The unimolecular dynamics of this I⁻‐‐‐CH₃I complex are very similar to those observed in the trajectory study, described above, of the Cl⁻‐‐‐CH₃Cl complex. This is expected, since both complexes are formed by ion–molecule association, and the I⁻ + CH₃I and Cl⁻ + CH₃Cl intermolecular potentials are very similar.

Non-RRKM dynamics and kinetics are found for the Cl⁻ + CH₃Br → ClCH₃ + Br⁻ S_N2 reaction [49–51]. For this reaction, the [Cl‐‐CH₃‐‐Br]⁻ central barrier TS is not rate controlling, and the kinetics of the Cl⁻‐‐‐CH₃Br pre-reaction complex must be modelled to calculate the S_N2 reaction rate constant versus temperature T or reactant relative translational energy E_rel. For the statistical model, RRKM theory is used to model the complex's kinetics, and this approach does not give rate constants versus either T or E_rel which agree with experiment. Experimental studies of the Cl⁻‐‐‐CH₃Br complex have shown that its dynamics are non-RRKM [52]. When this complex is excited by a low-power continuous-wave CO₂ laser at 943 cm⁻¹, it only decomposes to ClCH₃ + Br⁻. By contrast, RRKM theory predicts that this excited complex should also decompose to Cl⁻ + CH₃Br. The mode excited by the 943 cm⁻¹ radiation is an intramolecular CH₃ rocking mode. That only the ClCH₃ + Br-products are observed is consistent with slow IVR between the intramolecular and intermolecular modes of the Cl⁻‐‐‐CH₃Br complex. Decomposition to form Cl⁻ + CH₃Br requires energy transfer from the initially excited CH₃ rock intramolecular mode to the Cl⁻‐‐‐C stretch intermolecular mode.

In addition to the above S_N2 studies, there are other chemical dynamics simulations which have found extensive non-statistical dynamics for organic reactions [42]. Non-RRKM dynamics has been observed in both experiments [53] and simulations [54] for the isomerization of the acetone-enol cation CH₃–COH–CH₂⁺ to the acetone cation CH₃–CO–CH₃⁺, which then dissociates to CH₃⁺ + CH₃CO and to CH₃CO⁺ + CH₃. What is found is that the methyl group formed by this isomerization is 1.36 ± 0.15 times more likely to dissociate from the acetone cation, while RRKM theory predicts equal dissociation probabilities for the two −CH₃ groups.

Non-RRKM and non-statistical dynamics were found in chemical dynamics simulations of the ozonolysis of vinyl ethers, the results of which explain experimental observations [55]. The unimolecular dynamics of the primary ozonide, formed by the 1,3-dipolar addition of ozone to the vinyl ether, is non-statistical. The unimolecular kinetics of the ozonide was determined both experimentally and computationally, versus the size of the alkyl group of the vinyl ether, and the effect of the alkyl group size was found to be much less than the prediction of RRKM theory. IVR for the primary ozonide is incomplete and does not involve all of its vibrational modes on the time scale of its unimolecular decomposition. These results are consistent with the non-RRKM behaviour found for the simulations of the O₃ + propene unimolecular dynamics [9].

(c). Non-intrinsic reaction coordinate dynamics

The above PES bifurcation, F⁻ + CH₃OH, F⁻ + CH₃OOH and OH⁻(H₂O) and F⁻(H₂O) + CH₃I dynamics [20–25,37,38,40,41] are examples of non-IRC dynamics. The IRC for the PES bifurcation leads to the VRI point and not to reaction products. The IRC for the F⁻ + CH₃OH reaction leads to the CH₃OH‐‐‐F⁻ post-reaction complex and formation of this complex is unimportant in the reaction dynamics. For the F⁻ + CH₃OOH reaction, the IRC leads to the CH₂(OH)₂‐‐‐F⁻ complex and none of the randomly sampled trajectories form this complex. Such non-IRC dynamics are also found for cyclopropyl radical ring opening [56], 1,2,6-heptatriene rearrangement [57], the heterolysis rearrangement of protonated pinacolyl alcohol [58] and in the H + HBr → H₂ + Br and O(³P) + CH₃ → H₂ + H + CO reactions [59,60]. Non-IRC dynamics may be important for the ‘roaming mechanism’ [61], which is discussed below, and non-IRC H-atom roaming was found in earlier studies of H-atom transfer [62,63]. Non-IRC dynamics is an important property of many chemical reactions.

(d). Bimolecular reactions

(i). Non-statistical dynamics

The statistical model assumes that the chemical reactants are trapped in pre-reaction complexes on the PES. However, chemical dynamics simulations show that S_N2 nucleophilic substitution reactions, such as Cl⁻ + CH₃Cl → ClCH₃ + Cl⁻ and Cl⁻ + CH₃Br → ClCH₃ + Br⁻, often occur by direct mechanisms without forming either the pre-reaction or the post-reaction ion-dipole complexes [64,65]; e.g. Cl⁻‐‐‐CH₃Br and ClCH₃‐‐‐Br⁻. For the Cl⁻ + CH₃Cl reaction, exciting the C–Cl stretch mode of CH₃Cl enhances the direct reaction mechanism [66]. The dominant mechanism for the S_N2 reaction may change from indirect to direct over a small change in the reactant relative translational energy E_rel. For the Cl⁻ + CH₃Br reaction, this crossover occurs at E_rel of approximately 6 kcal mol⁻¹ [65]. In comparison, the well depth for the Cl⁻‐‐‐CH₃Br entrance-channel complex is approximately 10 kcal mol⁻¹. There is a similar change from a dominant indirect to direct mechanism for the Cl⁻ + CH₃I S_N2 reaction [67], whose potential energy curve is shown in figure 1. The well depth for the Cl⁻‐‐‐CH₃I complex is 11.6 kcal mol⁻¹. For E_rel of 4.6 kcal mol⁻¹, the indirect mechanism contributes 83% and 70% of the S_N2 reaction as found from MP2/ECP/d and BHandH/ECP/d direct dynamics simulations, respectively. At the higher E_rel of 9.0 kcal mol⁻¹, only 1% and 12% of the reaction is indirect, as found by these two respective direct dynamics methods.

Direct reaction mechanisms are also important for the microsolvated reaction OH⁻(H₂O) + CH₃I [40], whose potential energy curve is shown in figure 6. At the low collision energy of 0.05 eV (1.2 kcal mol⁻¹), the reactants do not have sufficient energy to form the proton transfer products and only the S_N2 pathways are open. As found from a B97-1/ECP/d direct dynamics simulation, there is no trapping in a post-reaction complex and 23% of the reaction occurs via a direct mechanism. Seventy-seven per cent of the trajectories have an indirect mechanism and are temporarily trapped in the (H₂O)HO⁻‐‐‐HCH₂I and/or CH₂I⁻‐‐‐H₂O(H₂O) pre-reaction complex, as assumed by the statistical model.

In contrast to the prediction of RRKM and transition state theories, chemical dynamics simulations show that central barrier recrossing is important for the Cl⁻ + CH₃Cl and Cl⁻ + CH₃Br S_N2 reactions [68,69]. These dynamics were studied for the Cl⁻ + CH₃Cl S_N2 reaction, first with an analytic PES [68] and then by MP2/6-31G* direct dynamics [70]. Trajectories were initiated at the [Cl‐‐CH₃‐‐Cl]⁻ central barrier with a 300 K Boltzmann distribution of energy as assumed by TST and then integrated in both the reverse and forward directions. The dynamics predicted by RRKM theory is that the trajectories form the ion−dipole complexes in moving off the barrier (i.e. Cl⁻‐‐‐CH₃Cl for reverse and ClCH₃‐‐‐Cl⁻ for forward) and then these complexes dissociate to reactants or products. The statistical kinetics assumes that crossing the barrier is rate controlling with no recrossings. The trajectories for the analytic PES and direct dynamics were integrated for 40 and 3 ps, respectively. RRKM theory predicts that 100% and 50% of the complexes should dissociate, for these respective simulations. By contrast, none of the complexes dissociated and the trajectories were ‘dynamically trapped’ in a region of the phase space about the [Cl‐‐CH₃‐‐Cl]⁻ central barrier with extensive central barrier recrossings [68,70].

To quantify the importance of the central barrier recrossings, the number of trajectories residing in the post-reaction ClCH₃‐‐‐Cl⁻ complex when the trajectories terminated was divided by the number of Cl⁻‐‐‐CH₃Cl → ClCH₃‐‐‐Cl⁻ forward barrier crossings. This ratio is 0.1 for the analytic PES dynamics and 0.2 for the direct dynamics. Because none of the trajectories actually reached the ClCH₃ + Cl⁻ product asymptote, more recrossings are expected if the trajectories were integrated for a longer time, resulting in a smaller value for the above ratio which TST assumes to be unity.

Central barrier recrossings are much less important for the Cl⁻ + CD₃Cl and Cl⁻ + C₂H₅Cl S_N2 reactions [71,72]. Trajectories were initiated at the [Cl‐‐CD₃‐‐Cl]⁻ central barrier [71] and integrated in both the forward and reverse directions for 3 ps as for the [Cl‐‐CH₃‐‐Cl]⁻ central barrier trajectories. Central barrier recrossings are much less important for Cl⁻ + CD₃Cl and the above reaction to barrier recrossing ratio is 0.7 when compared with 0.2 for Cl⁻ + CH₃Cl reaction. This difference may result from the lower intramolecular vibrational frequencies for CD₃Cl when compared with CH₃Cl, giving rise to more couplings between the low- and high-frequency vibrational modes of the reactive system and less phase space trapping in the vicinity of the central barrier. For the Cl⁻ + C₂H₅Cl S_N2 reaction, there is negligible recrossing of the central barrier [72]. The majority of the trajectories move off the central barrier to form the Cl⁻‐‐‐C₂H₅Cl complex and undergo efficient IVR and statistical/RRKM unimolecular dynamics to form Cl⁻ + C₂H₅Cl. However, a small number of the trajectories move directly to products without forming a complex, a non-RRKM result. The statistical dynamics for the Cl⁻‐‐‐C₂H₅Cl complex my result from the low-frequency C₂H₅ torsion and bending modes which couple with the complex's intermolecular modes and facilitate IVR.

(ii). Reactant reactive states

As discussed above for C₂H₅F → HF + C₂H₄ dissociation, in moving from the TS to products, the product energy states (i.e. translation, rotation and vibration) are not populated in a statistical manner. Similar dynamics, with time reversal, are expected in moving from the TS to reactants [73]. These dynamics will identify the probabilities for the different reactant states to form products. The reaction is not statistical in the sense that the reactant states do not have identical probabilities for forming products.

For triatomic A + BC reactions, it is well understood whether translation or vibration excitation of the reactants promote reaction, which depends upon the position of the TS on the PES; i.e. early for translation and late for vibration. These have been identified as the ‘Polanyi rules’ [74]. These rules are not easily transferable to polyatomic bimolecular reactions [75–78]. The state-specific dynamics of these reactions have been considered and analytic procedures have been described for determining their state specificities [77,78].

Xu et al. performed a B3LYP/6-31G* direct dynamics simulation [79], similar to the direct dynamics for C₂H₅F dissociation, except the trajectories were directed towards the reactants. Their simulation determined which reactant vibrational states have a high probability of attaining the TS structure and forming products for 1,3-dipolar addition reactions of diazonium betaines to acetylene and ethylene. The trajectories showed that reactant translation supplies the largest amount of energy needed to reach the TS, but reaction cannot proceed without a large amount of vibrational excitation in the bending mode of the betaine (i.e. N₂O, N₃H and CH₂N₂).

Analyses and calculations for enzyme catalysis reactions have suggested that protein dynamics may be important for some of these reactions [80,81]. Specific vibrations may promote the reaction on a short time scale, while particular conformational motions may be important for longer time dynamics. It may be possible to investigate each of these possible dynamical attributes by chemical dynamics simulations initiated at the TS and then directed towards the reactants. It may be possible to perform these simulations by QM/MM direct dynamics and/or MM chemical dynamics.

If the different reactant states have dissimilar probabilities for accessing the TS and forming products, one may say that these dynamics are non-statistical. However, this does not lead to a violation of TST. The validity of TST requires that: (i) a Boltzmann distribution is maintained for the translational, rotational and vibrational energies of the reactants during the reaction and (ii) once a trajectory crosses the TS in the direction reactants → products, the trajectory does not ‘turn around’ and recross [1,82]. Thus, the reactants → products flux may be taken as the reaction rate. If the reaction is dominated by high probabilities for a small number of reactant states, maintenance of a Boltzmann distribution for these states may become an important matter. In this manner, strong non-statistical bimolecular reaction dynamics might lead to non-TST kinetics.

(e). Roaming dynamics

There may be important non-statistical dynamics as part of the roaming mechanism for chemical reactions [61]. The term ‘roaming’ was coined in 2004 as part of a detailed discussion of the dynamics for H₂CO → H₂ + CO dissociation [83]. Initial descriptions of H₂CO dissociation identified two pathways: i.e. the elimination pathway via a three-centred TS to form H₂ + CO and a bond dissociation pathway to form the radicals H + HCO (figure 8). From chemical dynamics simulations, and in interpreting experimental measurements, a roaming mechanism was identified in which the loosely bound H-atom for H‐‐‐HCO dissociation ‘roamed’ around HCO and ultimately abstracted the H-atom from HCO forming H₂ + CO. For an H + HCO collision, this would be identified as a disproportionation reaction [84]. For H₂CO dissociation, roaming clearly couples the molecular elimination and bond dissociation pathways [61].

Figure 8.

Schematics of a roaming trajectory in the dissociation of H₂CO that initially moves in the direction of the radical H + HCO products, but turns in the high-energy region of the PES towards the high-energy isomers HOCH and ultimately undergoes an abstraction reaction to form H₂ + CO. Panel (a) shows the energy profile of the PES and a cartoon of the roaming pathway. Panel (b) is a perspective plot of the PES illustrating the ‘turn’ made from the radical pathway towards the isomers. Adapted from [61].

Roaming has now been reported for numerous reactions, as described in [83]. Extensive analyses of roaming have been reported for CH₃CHO [85–87] and NO₃ [88–90] dissociation and isomerization in CH₃NO₂ [91–93]. Roaming is expected to be important for hydrocarbon dissociation [94] and for C₃H₈ → CH₃ + C₂H₅ may lead to the possible disproportionation products CH₄ + C₂H₄ and CH₂ + C₂H₆. Roaming for this reaction has a weak CH₃‐‐‐C₂H₅ van der Waals interaction with possible minima and TSs in the roaming region of the PES.

The nature of the roaming dynamics, i.e. whether statistical or non-statistical, has received considerable attention [61]. A statistical theory for the kinetics and dynamics of roaming reactions has been proposed [95]. For roaming in formaldehyde, a statistical model may propose a TS for H‐‐‐HCO dissociation entering the roaming region of the PES and then a TS for H-atom abstraction to form H₂ + CO. This statistical model would assume rapid IVR within the roaming region and RRKM kinetics for leaving this region. A statistical-like kinetic model has been proposed for the formation of and reaction from the formaldehyde roaming region, and rate constants for this model were determined from trajectory data [96].

Mauguière et al. [97–99] have used a two-dimensional model, developed from the formaldehyde ab initio potential of Bowman and co-workers [100], to understand roaming dynamics for formaldehyde. They find that a phase space structure analysis of the roaming region is needed to understand the roaming dynamics [99]. Periodic orbit dividing surfaces are needed to delineate the different dynamics within the roaming region [101,102]. From the analysis of the phase structure for their model, they conclude that the roaming dynamics of formaldehyde are non-statistical and TST is not applicable.

A possible shortcoming of the phase space structure analysis for the formaldehyde roaming dynamics is that the Hamiltonian is a two-dimensional model and does not include all three H‐‐‐HCO intermolecular degrees of freedom. However, it is possible that the addition of the third intermolecular degree of freedom will not significantly alter the non-statistical dynamics found for roaming in formaldehyde. The HCO intramolecular degrees of freedom are expected to be only weakly coupled to the H‐‐‐HCO intermolecular modes. On the other hand, overall rotation and vibrational/rotational coupling could conceivably be important for the roaming intermolecular dynamics. Houston et al. [96] have studied formaldehyde trajectories using a full-dimensional potential. Azimuthal rotation of the roaming hydrogen atom about the CO bond axis of HCO was found to be a principal feature of the roaming. Roaming dynamics of reactive systems with more atoms may have low-frequency intramolecular modes which would couple to the intermolecular modes. Additional phase space structure analyses for roaming dynamics are expected.

3. Summary and future studies

The simulation and experimental studies described above illustrate that there are multiple non-statistical dynamics often observed for chemical reactions. Though simulation and theoretical studies of model systems may provide significant insight in non-statistical reaction dynamics, there is a sense that a deeper and more relevant understanding of the non-statistical dynamics may be obtained when both simulation and experiment tackle a particular problem. Examples of this are the studies for roaming in formaldehyde [61,96] and the dynamics of S_N2 reactions [103].

What would be particularly meaningful is the development of unifying theoretical models for non-statistical dynamics. For statistical reaction dynamics, the RRKM and TS theories unify disparate experimental systems and provide a means for their comparisons. To illustrate important issues for non-statistical dynamics, consider non-IRC dynamics which avoid exit-channel potential minima on PESs [37,38] and, thus, form non-IRC reaction products [38]. May a theoretical model be developed which would predict when and for which reactions such dynamics are expected? For example, it may be possible to develop a theoretical model which relates non-IRC dynamics with strong curvature along the IRC reaction path [38,104].

Many of the above simulations involve direct dynamics [105]. This is a general approach for studying atomistic reaction dynamics, without the need for developing an analytic PES. As a means to interpret experiments and understand their non-statistical dynamics, many more direct dynamics simulations are expected. However, it is important to ensure that the electronic structure theory used for the direct dynamics gives accurate atomistic dynamics for the chemical reaction under investigation.

Competing interests

We declare we have no competing interests.

Funding

The atomistic simulations for the Hase research group are supported by the National Science Foundation under grant no. CHE-1416428, the Robert A. Welch Foundation under grant no. D-0005 and the Air Force Office of Scientific Research under AFOSR Award no. FA9550-16-1-0133.