Solvation Dynamics and the Nature of Reaction Barriers and Ion-Pair Intermediates in Carbocation Reactions

Abstract

Additions of acids to 1,3-dienes are conventionally understood as involving discrete intermediates that undergo an ordinary competition between subsequent pathways to form the observed products. The combined experimental, computational, and dynamic trajectory study here suggests that this view is incorrect, and that solvation dynamics plays a critical role in the mechanism. While implicit solvent models were inadequate, QM/QM’ trajectories in explicit solvent provide an accurate prediction of the experimental selectivity in the addition of HCl to 1,3-pentadiene. Trajectories initiated from a protonation saddle point on the potential of mean force surface are predominantly unproductive due to a gating effect of solvation that allows diene protonation only when the incipient ion pair is neither too solvent-stabilized nor too little. Protonation then leads to relatively unsolvated ion pairs, and a majority of these collapse rapidly to the 1,2-product, without barrier and without achieving equilibrium solvation as intermediates. The remainder decay slowly, at a rate consistent with equilibrium solvation as true intermediates, affording a mixture of addition products. Overall, an accurate description of the nature and pathway selectivity of the ion pair intermediates in carbocation reactions must allow for species lacking equilibrium solvation. Potential reinterpretations of a series of historically notable observations in carbocation reactions are discussed.

Graphical Abstract

INTRODUCTION

Chemical reactions are understood from their mechanism. Mechanisms in solution are generally viewed as completely defined by a sequence of transition states and intermediates connecting the reactants to the products. The centrality of the transition state/intermediate idea to mechanistic chemistry has arguably grown over time owing to the focus of computational mechanistic studies on such structures. In recent years, however, there has been an explosion in the recognition of reactions that are not describable so simply. The regular mechanistic paradigm fails when the understanding of experimental observations requires the consideration of additional information, such as the multifurcating shape of an energy surface,^1–4 the excess energy in an intermediate and its distribution,^5–8 or the nonequilibrium solvation of structures.⁹ We describe all such reactions as involving “dynamic effects” in that they possess experimental characteristics that can be understood by allowance for the motions and momenta of atoms, if not for now by the standard statistical theories of chemistry.

The common denominator for dynamic effects is time. That is, when an experimental observation is associated with the evolution of a sufficiently short-lived structure, whether it is a formal intermediate or simply on the slope of an energy surface, then the inaccuracy or inapplicability of statistical theories is unsurprising. Molecular dynamics is inherently nonlinear, and so chaotic, but the descent into the past-forgetting chaos that engenders statisticality takes time, often multiple picoseconds.

The short lifetimes of most carbocation intermediates¹⁰ make them ripe for dynamic effects.¹¹ The abundant observations in classical carbocation chemistry have usually been fit, by implicit assumption, into a normal mechanistic model in which carbocations and ion pairs act as ordinary intermediates and their reactions are controlled by the barrier heights along exit channels. In some carbocation reactions, however, the observations are more difficult to shoehorn into a standard mechanism. We explore such a case here involving a textbook observation in electrophilic addition reactions.

Carbocations are usually generated as mechanistic intermediates from neutral reactants in polar solutions. Transition states forming carbocations then involve some degree of charge separation, but not to the extent present in the resulting ion pair. The solvation of the transition state will be approximately at equilibrium for the partial charge separation present in the transition state.¹² After the transition state, charge separation continues as the reaction coordinate progresses. Progress along a reaction coordinate is fast, typically requiring on the order of 50–200 fs. The solvent adjusts to the continuing charge development in two ways, by the polarization of the nearby solvent molecules and by the reorientation of the solvent molecular dipoles. The former is relatively fast but the latter is slower, requiring from a few ps up to several hundred in typical polar organic solvents.^13,14 Our hypothesis is that the initial solvation of carbocation ion pairs is not at equilibrium and that this affects reactions that occur rapidly after carbocation formation. We describe here evidence to support this effect of nonequilibrium solvation in the addition of acids to 1,3-dienes. The results have broad implications toward the understanding of carbocation reactions and other polar reactions of short-lived intermediates.

Additions of acids to 1,3-dienes are the most common textbook examples of the complications associated with conjugation in intermediate ions. Two simple adducts may be formed as a result of either 1,2- or 1,4-addition (eq 1). The competition between the two addition modes involves a number of interesting issues, but we will focus here on the general observation that the kinetic selectivity favors the 1,2-addition product.^15,16 It was long recognized that the kinetic 1,2 preference could not be explained by the intermediacy of free allylic cations, but deconvoluting the role of multiple mechanistic complexities versus intrinsic structural preferences was difficult. A seemingly clear picture emerged when Nordlander et al. studied the addition of DCl to 1,3-pentadiene (1, eq 2).¹⁷ This reaction exhibits a substantial preference for the formation of the 1,2-adduct 2 over the 1,4-adduct 3. If the intermediate were either a free cation or an effectively symmetrical ion pair, then an equal mixture of 2 and 3 would be formed. The preference for 2 over 3 was proposed to result from the initial formation of an unsymmetrical ion pair (4-2Cl) that competitively either collapsed to the 1,2-product 2 or isomerized to the isotopomeric ion pair 4-4Cl. The latter could then afford the minor product 3. Nordlander’s idea of competitively reacting and isomerizing ion pairs has become the standard explanation for the 1,2 preference in these reactions. We will see evidence that this picture is not correct. Our results support a more interesting explanation, involving solvation dynamics.

(1)

(2)

From a wider perspective, the importance of ion pairs in carbocation reactions has long been recognized, and their nature was historically probed by many elegant experiments.^18–22 Our results will support an addition to the classical Winstein ion-pair framework and suggest some possible reinterpretations of classically important experimental observations.

RESULTS AND DISCUSSION

A Brief Digression: Carbocation or Concerted Ad_E3?

The particular reaction of 1/HCl in nitromethane as solvent was chosen for study, in part because of the relative simplicity of the system, and in part because the relatively high polarity of the nitromethane (∈ = 35.9) makes its role in the reaction more interesting and relevant to other polar reactions in solution. HCl is predominantly molecular at ordinary concentrations in nitromethane,²³ and the kinetics of its additions are first order in alkene and second order in HCl.^24,16 The overall termolecular kinetics might in general arise in two ways, with the second HCl molecule either accelerating the reaction as a nucleophile in a concerted Ad_E3 addition, or alternatively accelerating the reaction electrophilically by coordinatively increasing the acidity of the protonating HCl molecule in an otherwise ordinary carbocation mechanism. Nordlander did not discuss the Ad_E3 mechanism as a possibility, probably because the general expectation from the literature is that the concerted Ad_E3 mechanism occurs only when the carbocation would be highly unstable. For example, the reaction of HCl with 3-hexyne is an anti addition, presumably occurring by an Ad_E3 process, while the parallel addition to 1-phenylpropyne is syn and is thought to involve a carbocation.²⁵ For additions of HCl to simple alkenes such as isobutene in nitromethane, it is observed that free chloride suppresses the rate,²⁴ the opposite of observations in Ad_E3 mechanisms.^25a Also, the addition of HCl to cis- and trans-1-phenylpropene in nitromethane affords the syn-addition adduct as the major product.²⁶ These observations weigh strongly against a concerted Ad_E3 mechanism.

In contrast to the observations above and general expectations, one striking literature observation appeared to strongly support the Ad_E3 mechanism. That is, Pocker and Stevens reported in 1969 that the addition of DCl to 1-methylcyclopentene (5) afforded 96 ± 4% anti (6) over syn (7) addition.²⁷ This result is arguably the strongest evidence for an Ad_E3 mechanism in an alkene that could alternatively produce a tertiary cation, and it was seemingly the least ambiguous stereochemical evidence for an Ad_E3 addition to any alkene. (Additions to cyclohexene²⁸ exhibit anti addition but the significance of this is compromised by the asymmetry of the faces of a chair cyclohexyl cation.)

However, the anti stereoselectivity reported by Pocker and Stevens is incorrect. At the time, it was not possible to directly analyze the stereochemistry of the 6/7 mixture, and the stereochemical determination had been based on a mass spectrometry analysis of the alkene produced by treatment of the product mixture with base. This required a series of assumptions about the stereochemistry, regiochemistry, and isotope effect for the elimination. With the advantage of modern NMR methodology, we were able to directly examine the 6/7 mixture. The key observation (Figure 1) is that the mixture exhibits two deuterium-split triplets that are shifted upfield from the otherwise equivalent carbon by ~0.37 ppm.²⁹ The resolved triplets for the diastereomeric isotopomers were assigned from the mixture’s 2D HSQC spectrum (see the Supporting Information, SI), which show that the larger of the triplets correlates with the upfield (H trans to Cl) ¹H at δ 1.73, while the smaller triplet correlates to the downfield (H cis to Cl) ¹H at δ 2.12. The syn addition of HCl is thus the major pathway, by a ratio of 75:25 versus the alternative anti addition. This is inconsistent with a concerted Ad_E3 mechanism. Our results below will provide an alternative dynamical explanation for the observed syn selectivity.

Figure 1.

Expansion of the ¹³C NMR spectrum of 6/7 obtained from the reaction of 1-methylcyclopentene with DCl in nitromethane, showing the area for the D-labeled β carbon. The two small peaks at the left of the spectrum are ¹³C–¹³C satellites for the unlabeled β′ carbon.

Computational Methods Selection.

DFT methods tend to overestimate the proton affinity of 1,3-pentadiene in comparison with CCSD(T)/aug-cc-pVTZ calculations, most by 6–11 kcal/mol, but M06-2X calculations mirrored the ab initio value within 1 kcal/mol with little dependence on the basis set (see the SI). The proton affinities of Cl⁻ and [ClHCl]⁻ were underestimated by every DFT method explored, by 1–6 kcal/mol and 6–11 kcal/mol, respectively. In contrast, MP2/6-31+G** calculations performed uniformly well, within 1.5 kcal/mol. The MP2/6-31+G** calculations also closely reproduced DLPNO–CCSD(T)/aug-cc-pVTZ energies along a series of 98 points sampled from solution trajectories (see below), with a correlation R² of 0.996 and a slope of 0.98. M06-2X/6-31+G* calculations performed less well on each of these tests (see the SI), but were more practical for large-scale trajectory calculations in explicit solvent. In practice, both methods were fully explored and gave similar results. We will emphasize the MP2 calculations, since they were notably superior in every comparison with high-level energies from highly correlated methods.

It was anticipated that implicit solvent models would prove inadequate for this reaction, but they provide a baseline for consideration of the reaction in explicit solvent. Four series of stationary points and trajectories were obtained, arising from the use of the MP2 and M06-2X methods in combination with PCM and SMD implicit solvent models. Calculated free energies for stationary points are based on DLPNO–CCSD(T)/aug-cc-pvtz gas-phase potential energies, with PCM or SMD implicit solvent corrections, and MP2 or M06-2X enthalpies and entropies corrected to a 1 M standard state. The DLPNO–CCSD(T)//MP2/PCM results are presented here, but very similar results were obtained with all of the combinations. (See the SI for complete details and energies.)

For trajectories and free-energy surface calculations in explicit solvent, the computational model consisted of 1,3-pentadiene and either one or two molecules of HCl in a sphere of 120 nitromethane molecules with a diameter of 29.5 Å and a density of 1.18 g/mL. This model was explored on the QM/QM’ ONIOM surface, using either MP2/6-31+G** or M06-2X/6-31+G* for the 1,3-pentadiene and HCl, and using PM6-DH+ for the nitromethane. The low-level component of the ONIOM calculation was supplemented by an additional empirical dispersion term, modeled after the Grimme GD2 dispersion,³⁰ with a global parametrization to approximate a realistic density and heat of vaporization. A notable limitation of this approach is that the polarizability of the nitromethane is 29% too low, as is normal for semiempirical methods. The PM6-DH+ dipole moment for nitromethane is 3.98, in reasonable agreement with the experimental value of 3.46.³¹ The low solvent sensitivity of the Nordlander results suggested that the accuracy of the computational solvent may be adequate, but this will be probed by comparison of computed and experimental observations.

Inadequacy of Implicit Solvation.

Transition structures were located in the various calculations for the protonation of 1,3-pentadiene by either one or two molecules of HCl. The lowest-energy structure 8° is predicted to have a ΔG^‡ of 21.7 kcal/mol at 25 °C and a ΔH^‡ of only 9.1 kcal/mol. These barriers are consistent with the observation that these reactions occur experimentally at 25 °C. The implicit-solvent calculations thus appear satisfactory for the prediction of the barrier for the addition. However, they fail to account for the observed preponderance of the 1,2-addition product 2. Two approaches were used to gauge the products from each transition structure. The first was based on the dynamic reaction path (DRP) method,³² with the variation that the DRP was calculated repeatedly with the addition of a Boltzmann-random amount of energy to the transition vector ν^‡. In the case of 8^‡, the outcome of the DRPs varied with the energy put into ν^‡ with 86% 1,4-product and 14% 1,2-product. The alternative transition structure 9^‡ is only modestly higher in energy and would contribute significantly to the product mixture. The DRPs from this structure lead consistently to the 1,4-product 3. Neither prediction fits with the experimental preponderance of 2.

For a more complete evaluation of the products that would be produced from 8^‡ and 9^‡ within the implicit solvent model, each was used as the starting point for quasiclassical direct-dynamics trajectories on the MP2/6-31+G**/PCM surface. Each normal mode in the two structures was given it zero-point energy (ZPE) plus a randomized excitation energy based on a Boltzmann distribution, along with a randomized displacement of the modes. The trajectories were then propagated forward and backward in time until 2 or 3 was formed, the 4+/ClHCl⁻ ion pair dissociated (gauged by a minimum C–Cl distance of 5.0 Å), the starting materials were reformed, or a 500 fs time limit was reached. The outcome of these trajectories is summarized in Figure 2. In each case, dissociation of the ions predominated. When a neutral product was formed, it was predominantly the 1,4-product. Clearly, the implicit-solvent approach cannot account for experimental observations. Equivalent results were obtained with the SMD solvent model and in M06-2X/6-31+G** calculations with both the PCM and SMD solvent models (see the SI). As we observed previously in a study of the nitration of toluene,^9b implicit solvent models appear simply inadequate for the prediction of product mixtures that arise from the dynamics of ionic reactions in solution.

Figure 2.

Transition structures and product summaries for the addition of HCl to 1,3-pentadiene in implicit nitromethane. The barriers listed are DLPNO–CCSD(T)/PCM//MP2/6-31+G**/PCM with a 1 M standard state, at 25 °C, in kcal/mol.

Potential of Mean Force (PMF) Calculations in Explicit Solvent.

For the reaction of 1 with dimeric HCl in 120 nitromethane molecules, the 2-dimensional PMF surface for diene protonation along the C₁—H and H–Cl distance dimensions was determined by umbrella sampling in molecular dynamics (MD) calculations over a total of 1.0 ns. The PMF was then calculated by the weighted histogram analysis method.³³ The resulting surface is shown in Figure 3; similar surfaces obtained in M06-2X calculations and for protonation by monomeric HCl are shown in the SI. A saddle point 10^‡ on this surface has C₁—H and H–Cl distances of approximately 1.29 and 1.68 Å, respectively. The saddle point in explicit solvent is then in reasonable agreement with the implicit-solvent transition structures 8^‡ and 9^‡. By adding the free-energy cost of bringing the reactants together to form a prereactive complex in MP2/PCM calculations (6.8 kcal/mol) to the PMF barrier for reaching the saddle point (13.4 kcal/mol) from the area of this complex, a total reaction barrier of 20.2 kcal/mol may be estimated. This barrier is then in good agreement with the implicit solvent calculations and experimental observations.

Figure 3.

QM/QM′ PMF surface for protonation of 1,3-pentadiene by (HCl)₂ in explicit nitromethane. The energy scale is relative to a flat region of the surface in the area of a prereactive complex.

By its nature, a PMF surface of this type is a dimensionally trivialized representation of a high-dimensional problem.³⁴ In this reaction, the uncontrolled dimensions include both the motions of the second HCl molecule and the multitude of solvent motions associated with the protonation. The PMF saddle point then presents an incomplete and potentially misleading picture of the reaction. Our analysis below of trajectories started from the area of 10^‡ will examine the impact of these additional dimensions on the reaction.

Trajectories in Explicit Solvent.

The protonation of the diene and the subsequent 1,2- versus 1,4-product selectivity were explored in trajectory studies using the same 1,3-pentadiene/2-HCl/120 nitromethane system as that used for the PMF surface. The starting points for trajectories were obtained from a series of independent simulations that were equilibrated at 25 °C with the C₁—H and H–Cl distances constrained in the area of 10^‡ by a harmonic potential. At intervals of 250 fs, structures and velocities were extracted from the equilibrating systems and integrated forward and backward in time with no constraint. For clarity later, we define “forward in time” as the direction in which the transferring proton moves closer to C₁ of the diene. The trajectories were then stopped when the products 2 or 3 were formed, the starting materials were reformed, a time limit of 3000 fs was reached, or the Cl–HCl anion/pentadienyl cation tight ion pair dissociated. Dissociation was defined by the chlorides being >5 Å separated from the cationic carbons. The 5 Å criterion was chosen to delineate approximately the formation of a solvent-separated ion pair.³⁵

The trajectories of interest for understanding the rate and selectivity in this reaction are those that fully traverse the protonation transition state, passing from the starting materials to form the pentadienyl cation. We refer to such trajectories as “productive.” An expected but important observation for later analysis is that only a small proportion of the trajectories (22%) started from the area of 10^‡ are productive. The remainder are recrossing trajectories, passing either from starting materials to starting materials (“SM–SM”, 46%) or between the differing outcomes (2, 3, ion dissociation, time-limited ion pair) on the product-side of 10^‡ (“product—product”, 32%).

Comparison with Experiment. Redetermination of the Product Ratio.

A critical issue was whether the explicit-solvent trajectories could account for the experimental product ratio. The productive trajectories include those forming the 1,2- and 1,4-products with either of the two chloride anions along with trajectories that result in dissociation of the ion pair and those reaching the time limit as an associated tight ion pair. Table 1 summarizes the results from trajectories with either MP2 or M06-2X employed as the high-level component of the ONIOM calculations. To predict the product ratio from these results, we make the assumption that the dissociating and time-limited trajectories would ultimately form equal amounts of 1,2 and 1,4 products. The expectation that dissociated ions will reassociate randomly seems evident, but the disposition of the long-lived tight ion pairs will require further consideration below. Allowing for the mixture of products from these trajectories leads to predictions of 77% 1,2-product from the MP2 ONIOM simulation and 69% 1,2-product from the M06-2X ONIOM.

Table 1.

Selectivity in Additions of HCl to 1,3-Pentadiene in Explicit Nitromethane at 25 °C

	MP2 ONIOM	M06-2X ONIOM
% productive	22%	16%
1,2-product	265	94
1,4-product	5	13
dissociation of ions	77	66
ion pair reaches time limit	128	41
predicted% 1,2-producta	77% ± 2%	69% ± 3%
	experimental 1,2 selectivity 68% (Nordlander) 74% ± 2% (this work)

^aCalculated as the number of 1,2-product forming trajectories plus half of the dissociating and time-limited trajectories, divided by the total of productive trajectories.

Both predictions appeared in reasonable agreement with Nordlander’s observation of 68% 1,2-product in nitromethane at 25 °C, but we were bothered that experiment agreed better with the computationally less defensible M06-2X calculations. Nordlander’s measurement appeared subject to possible error since it was based on incompletely resolved ²H NMR peaks and since the analysis required purified material subject to isomerization under the reaction conditions and in the isolation process. Accordingly, after all of the computations for this paper were complete, we undertook a redetermination of the product ratio. The deuterated carbons in 2 and 3 are completely resolved from their unlabeled isotopologs, and their ratio was readily determined by direct ¹³C NMR analysis of the reaction mixture without isolation, after quenching of the excess acid with 2-trimethylsiloxypropene. It was found that the products 2 and 3 equilibrate under the reaction conditions, albeit slowly. The most rapid practical analysis observed the products in a 73.5:26.5 ratio, with an uncertainty of ±1.6% arising from the limited signal-to-noise ratio possible for the reaction mixture analysis. Extrapolation of this ratio back to time zero finds that 74% ± 2% of the 1,2-product was present in the kinetic mixture.

The limited number of solvent molecules and the underestimation of their polarizability in the ONIOM method employed are significant limitations in the computational models. However, their ability to accurately account for the product selectivity supports the adequacy of the ONIOM models for understanding the origin of the 1,2-selectivity in these reactions.

The Multi-Stage Nature of Product Formation.

The productive trajectories exhibit two striking features. The first of these is that very few trajectories afford the 1,4-product within the 3000 fs time limit. Only five of the MP2 ONIOM trajectories form the 1,4-product, and two of these involve initial geometries that place the second HCl molecule in a position that is favorable for its chloride to attack C₄ of the diene, leading to product derived from the distal chloride. The low formation of 1,4-product might be understandable if 1,2- and 1,4-ion pairs rarely interconvert, but in fact ion-pair equilibration happens routinely. The ion pairs in trajectories that last to the time limit of 3000 fs undergo isomerization, defined by a switch in whether C₂ versus C₄ is closest to a chloride atom, an average of six times (median of four). This observation supports our assumption above that the time-limited ion pairs will afford equal amounts of 1,2 and 1,4 products. Equilibration, however, does not lead to much 1,4-product formation within 3000 fs. Instead, the trajectories that last long enough to undergo ion-pair equilibration rarely form either product at all!

This leads to the most intriguing observation for the trajectories, which is that product formation at first occurs rapidly then nearly stops. Half of all of the observed product formation occurs within the first 640 fs, 75% within 1000 fs, and 90% within 1500 fs (Figure 4). Although 50% of the ion pairs survive to 1000 fs, any subsequent product formation is slow and progressively slower, with dissociation of the ions becoming their major decay route.

Figure 4.

Decay of the ion pairs versus time after release from 10^‡, along with an exponential fit to the first 800 fs and a histogram of the timing of 1,2-additions. The rapid early decay predominantly results from formation of the 1,2-addition product, while dissociation is the major process in the slow late decay.

The plot of the fraction of surviving ion pairs versus time after release from the area of 10^‡ in Figure 4 can be divided into three phases. The first is a lag time of approximately 200 fs, required for completion of the proton transfer (see the discussion later) and for the chloride of the HCl performing the protonation to approach C₂ of the diene from a median starting separation of ~3.6 Å, requiring a relative motion of ~1.8 Å. The second phase involves the rapid decay of ion pairs as chloride attack at C₂ occurs. Dissociation of ions is a minor pathway in this time from but it is notable that, like the 1,2-addition, the dissociation process also involves an initial rapid phase. The third phase of ion pair decay is slow, far slower than the illustrated exponential decay fit to the first 800 fs. A double-exponential fit of the data found nominal rate constants of 1.8 × 10¹² s⁻¹ (± ~ 15%) and 1.3 × 10¹¹ s⁻¹ (± ~ 30%), with the slow phase roughly 13 times slower than the rapid-decay phase. Qualitatively, the decay of the tight ion pairs is similar to what might be observed if the protonation step formed two distinct species that react at different rates.

To understand this dichotomous behavior of the trajectories, we examine in more detail the role of the solvent in these reactions. This starts with an examination of how the solvent impacts which trajectories are productive.

Solvation Dictates Productive Trajectories.

Only 22% of the trajectories initiated from 10^‡ were productive. The idealized expectation arising from the “no-recrossing” assumption of conventional transition state theory is that trajectories initiated from a transition state and propagated forward and backward in time will all be productive. In practice this does not happen for any modestly complex reaction, even for gas-phase simulations, but trajectory simulations of simple organic reactions started from a well-defined transition state often result in greater than 90% productive trajectories. An alternative possibility for consideration is that the outcome of the forward and backward branches of trajectories could be independent of each other. If the outcome of each branch were fully random, then ~50% would be productive (49% for the observed excess of SM-SM trajectories). The much lower proportion of productive trajectories observed here excludes a random outcome. Instead, there is specific preference for recrossing.

This is unsurprising, especially for a polar reaction in a polar solvent.^36,37 The starting points for trajectories have controlled only two dimensions of a high-dimensional transition state. The many uncontrolled dimensions when starting from 10^‡ can readily take on values that displace the trajectory starting points away from the actual transition states. The uncontrolled dimensions then serve as a gate that enforces recrossing most of the time but allows a productive trajectory when these dimensions have values in the area of a true high-dimensional transition state.

This idea is well established but vague. We therefore sought to uncover what aspects of the trajectory starting points dictated whether they were productive versus recrossing. We first explored whether dimensions internal to 10^‡, such as the orientation of the second HCl molecule or the particular starting distances a, b, c, and d affected the likelihood of a trajectory being productive. Within a broad range, the effect of these internal dimensions was minor (see the SI).

The initial orientation of the solvent however had a major effect on the productivity of trajectories and their outcome on the product-side of the transition state. To evaluate the role of solvent, we defined an “incipient ion-pair stabilization parameter” E_ip for each point in the trajectories. The E_ip value, defined by eq 3, represents the hypothetical electrostatic stabilization that would be present if the solvent atoms were atomic point charges (using equilibrium M06-2X/6-31+G**/MK charges, chosen based on dipole-moment accuracy) and if the solute atoms took on the charges calculated for separated pentenyl cation and [ClHCl]⁻ anion (see the SI for calculational procedures and congruent results obtained by alternative approaches). The E_ip then represents the degree to which the solvent would electrostatically stabilize the protonation of 1 to form an ion pair or the continued presence of the ion pair. An E_ip near zero would mean that the solvent is randomly oriented and unable to stabilize an ion pair, while an E_ip of approximately −72 kcal/mol is the maximum average stabilization of full ion pairs by the solvent. It should be noted that structures in the area of 10^‡ have taken on only approximately 55% of the charge separation present in the ultimate ion pair (based on MK charges for 8^‡). The average starting-point E_ip for all trajectories reflects this partial charge separation, being −39 kcal/mol or a little more than half of the maximum.

$$E_{\mathrm{ip}}=\sum _{\text{i,j}}\frac{q_i{q}_j}{r_{\mathrm{ij}}}$$ (3)

q_i = atomic charges in solute if fully ionized

q_j = atomic charges in solvent

A key observation is that the initial E_ip was associated strongly with the outcome of trajectories (Figure 5). For trajectories that underwent recrossing on the starting-material side of the transition state, the average initial E_ip was only −34.9 kcal/mol. This shows that the solvent was on average not well disposed to stabilize the charge separation that would result from proton transfer. Trajectories released from 10^‡ would take an initial step toward ion-pair formation but the proton transfer could not be completed in trajectories with a low initial E_ip.

Figure 5.

Plots of the average values for the incipient ion-pair stabilization parameter E_ip versus time for trajectories with differing outcomes. For each trajectory, the E_ip was only followed “forward in time”, defined as the direction in which the initial motion of the proton transfer was toward C₁ of the diene.

In contrast, trajectories that underwent recrossing on the product side of the transition state were associated with a high initial E_ip of −51.5 kcal/mol. In these cases the solvent is initially arranged to strongly stabilize charge separation, making it energetically disfavored for the trajectories to revert to the starting materials.

In either case, solvent reorganization simply takes too much time to allow the transition state to be crossed. In Hynes theory the relevant time scale is the longitudinal dielectric solvent relaxation time, τ_L.³⁸ The τ_L for nitromethane is ~400 fs at 25 °C,³⁹ and the average decay of E_ip in Figure 5 appears to reflect this reasonably.⁴⁰

The productive trajectories are associated with a “Goldilocks” level of initial electrostatic stabilization of charge separation, with E_ip averaging −42.2 kcal/mol. The middle range of E_ip appears to provide an energetic balance between reversion to starting material and completing protonation to form the ion pair. This allows trajectories to traverse from one realm to the other.

The time course of the E_ip followed forward in time from 10^‡ also bears an intriguing relationship with the choice of outcomes among productive trajectories. In trajectories that result in ion-pair dissociation, the E_ip rapidly drops to near its terminal value (Figure 5, purple line), indicating that the solvent has become organized to optimally stabilize the separating ions. The solvent organization appears to be a cause, rather than an effect of dissociation, because the greater E_ip shows up in the first 200 fs while the median time for dissociation is 1020 fs. Trajectories that ultimately form the 1,2-product in contrast exhibit a much smaller stabilization of the ion pair (Figure 5, blue line). In such trajectories, the solvent is never fully organized to stabilize an ion pair. The ions are formed in a relatively nonstabilizing “electrostatic hole” in the solvent, then spend their first few hundred femtoseconds pushed toward tight association by this low stabilization. Within 1000 fs, three-quarters of these ion pairs annihilate.

The curves in Figure 5 represent averages of large numbers of trajectories, while individual trajectories vary. The standard deviation for the initial E_ip in each category is ~7 kcal/mol. The categories then overlap, but the solvent orientation as gauged by E_ip is a strong predictor of individual trajectory outcomes. For example, a trajectory with a starting E_ip of −51.5 has a 33-fold greater chance of undergoing product-side recrossing than a trajectory with a starting E_ip of −34.9.

Another way in which the solvation influences the reaction in the area of the proton transfer can be seen when individual trajectories are examined. Figure 6 shows the early time course of the E_ip for a series of 1,2-product forming trajectories with completion times of ~700 fs. As the trajectories leave the area of the transition state, the solvation slowly changes in the direction of greater stabilization of the forming ion pair. The proton transfer itself, however, is remarkably slow. To assess this, we examined the time required for the bottom of the proton’s energy well, as judged by the forces on the proton, to be at a distance from C₁ that is less than 1.14 Å. (It would be misleading to examine the proton position by itself, since it vibrates broadly within its energy well.) Figure 6 shows these proton-transfer completion points on each trajectory, and the median time after release from the TS was ~50 fs. For comparison, an ordinary C–H vibration takes ~11 fs and moves a proton an equivalent distance within 6 fs. Completion of the proton transfer requires that the solvation first energetically allow it by reorganizing to stabilize the forming ion pair. This takes time, so the simple proton transfer is a surprisingly plodding process.

Figure 6.

Plots of the incipient ion-pair stabilization parameter E_ip versus time for individual productive trajectories.

Is There an Intermediate?

The term “intermediate” can be misleading when applied to the ion-pair structures that are formed after the proton transfer step. An intermediate is defined by IUPAC as a molecular entity in a mechanism that has a lifetime that is “appreciably longer than a molecular vibration” and corresponds to a “local potential energy minimum.”⁴¹ With minimum and median lifetimes of ~200 and ~1000 fs, the ion pairs are sufficiently long-lived to be considered intermediates, but whether they exist within energy minima is a more complicated problem on multiple levels. We have argued previously that free energy is more appropriate than potential energy as a defining feature for intermediates.^2e We will follow that definition here, and a species that exists for an extended time in a free-energy well will be described as a “true” intermediate. It should be recognized that free energy is an equilibrium concept that is not rigorously definable at sufficiently short time scales.

To explore the equilibrium free-energy landscape available to the 4⁺/ClHCl⁻ ion-pair structures, a one-dimensional PMF surface based on the C–Cl distance was determined by umbrella sampling in MD calculations over a total of ~1 ns. The complication that either C₂ or C₄ may undergo bond formation with either of the two chlorine atoms was handled by an interlocking-sphere algorithm,^9b (see the SI) so that the PMF obtained (Figure 7) applies to the shortest of the four relevant C—Cl distances. The PMF has a notable broad minimum at a C—Cl distance of ~4.1 Å. This supports the potential existence of a true intermediate in the system, which we will describe as the “solvent-equilibrated ion pair.” To form the neutral product, the solvent-equilibrated ion pair must surmount a small barrier in the PMF, only 0.5 kcal/mol, to pass through a transition state with a C—Cl distance of 3.5 Å. The ~20 kcal/mol barrier for formation of the solvent-equilibrated ion pair from product is notably in good agreement with the experimental observation that a slow product isomerization occurs at 25 °C.

Figure 7.

QM/QM′ PMF for the approach of a chloride ion to either C₂ or C₄ of 4⁺ in explicit nitromethane. The energy scale is relative to a minimum on the surface at 1.85 ± 0.025 Å.

Trajectory Behavior of Solvent-Equilibrated Ion Pairs.

The existence of a true intermediate in the system leaves open the question of whether addition pathways actually pass through the intermediate, or alternatively bypass it. To gauge this issue, we studied the behavior of trajectories initiated from the area of the solvent-equilibrated ion pair. The starting points for trajectories were obtained from a series of independent simulations that were equilibrated at 25 °C with the C—Cl distances constrained in the area of the intermediate by a harmonic potential. At intervals of 250 fs, structures and velocities were extracted from the equilibrating systems and integrated forward in time with no constraint. As shown in Figure 8, the resulting trajectories decayed slowly with time to a mixture of the addition product and dissociated ions, but 63% of the solvent-equilibrated ion pairs survive to the time limit of 3000 fs.

Figure 8.

Decay of solvent-equilibrated ion pairs, and an exponential fit.

The decay fit well with a single exponential with a rate constant of 1.7 × 10¹¹ s⁻¹. This may be compared with the 1.3 × 10¹¹ s⁻¹ observed above for the slower phase of the double-exponential decay of the original trajectories. Allowing for the uncertainty in each number, particularly the latter due to the double-exponential fitting, the two rate constants are indistinguishable. This agreement supports the idea that the slower phase of the decay of the ion pairs formed on protonation of 1 results from the reaction of species that approximate solvent-equilibrated ion pairs.

A Consistent Mechanism.

The observation in the last section, in combination with the strong and time-dependent effect of solvation in stabilizing the formal ion pairs, supports a mechanism that may be viewed as the combination of two limiting prototypes. At one extreme, the solvation-gated protonation of 1 through 10^‡ leads in time to solvent-equilibrated ion pairs. Such ion pairs slowly form the addition product in competition with dissociation, and they are representatively viewed as intermediates within the conventional ideas of mechanistic chemistry. These intermediates have time to geometrically equilibrate, or dissociate, and lead to a mixture of 1,2- and 1,4-addition products by passage over barriers.

At the opposite extreme, the protonation of 1 through 10^‡ leads to a formal ion pair that is neither fully solvated nor becomes so quickly. The prevalence of such “holes” in the solvation for ion pairs at their creation may be understood from two perspectives. The first is to recognize that the transition state for protonation is not as polar as the ion pair, so solvation that is approximately optimized for the protonation process will not be so for the resulting ion pair. The second perspective is that solvation that strongly stabilizes the ion pair promotes recrossing trajectories, not reactive trajectories. At this extreme, the ion pairs afford the product ~13 times faster than the solvent-equilibrated ion pairs. These ion pairs exist only on the slope of a notional free energy surface that leads inexorably to exclusively 1,2-product formation. This does not make the adduct formation instantaneous; as we have seen previously with nitration reactions, the trajectories of barrierless reactions can proceed slowly as they await solvent reorganization.^9b However, the absence of a barrier for their reaction, or equilibrated solvation as they decay, makes their consideration as conventional mechanistic intermediates misleading.

These extremes are likely an incomplete description. The initial E_ip for reactive trajectories is not bimodal, but rather approximates a normal distribution. Ion pairs are formed with a range of solvent stabilizations (with a σ of ~7 kcal/mol) but none have an initial E_ip that is very close to equilibrium. Then, as illustrated in Figure 6, all of the ion pairs tend to show an increase in their solvation over time, at varying rates. The ion-pair decay of Figure 4 was modeled using two exponentials that can be taken as representing the two limiting pathways, but this cannot be cleanly distinguished from a continuum.

A closely analogous mechanism would provide a consistent explanation for syn selectivity in the addition of DCl to 5. We would suggest that the protonation step can lead to, at one extreme, a weakly solvated ion pair that collapses without barrier to the major syn addition product 7, and, at the other extreme, a highly solvated ion pair that may dissociate or isomerize to afford a mixture of syn and anti addition products. In this regard, it is notable that the syn selectivity with 5 and the 1,2-selectivity with 1 are similar.

CONCLUSIONS AND PERSPECTIVE

The trajectory studies in explicit solvent are able to accurately account for the 1,2 versus 1,4 product selectivity in the addition of DCl to 1 in nitromethane. In this way, they provide an experimentally consistent model for the mechanism of the reaction, one that differs in fundamental ways from the conventional mechanism. The protonation of the diene is gated by the solvent: solute geometries that nominally appear to be in the area of the transition state mainly lead to recrossing trajectories unless the solvent’s ability to stabilize the incipient ion pair is neither too high nor too low.

After the protonation, the solvation of the resulting ion pairs is not at equilibrium, and equilibrium solvation is on average approached over the course of a few hundred femtoseconds. The ion pairs decay to the product by two limiting processes. The faster of these leads exclusively to the 1,2-addition product, and its rate is inconsistent with that seen after fully equilibrated solvation. The slower decay is at a rate that cannot be distinguished from that of equilibrated ion pairs, and this lower rate appears associated with a barrier that is observed on the PMF surface. These ion pairs lead to a mixture of 1,2 and 1,4 products.

These results suggest a modification of the classical Winstein formulation of the role of ion pairs in carbocation reactions. That is, the tight ion pairs that are produced in ordinary carbocation reactions are initially formed with nonequilibrium solvation, and these species may react differently from solvent-equilibrated ion pairs. Their partitioning between subsequent pathways in particular may not be decided by relative free-energy barriers. Whether or not such ion pairs are designated as “intermediates,” their evolution is akin to that of molecules on the slopes of bifurcating energy surfaces.^1–4 With time, these species may lead to tight ion pairs whose solvation is at equilibrium. It then becomes meaningful to describe the ion pair as having a well-defined free energy, and having well-defined rates and free-energy barriers for its subsequent reactions. As discussed in the introduction for other dynamic effects, the key factor distinguishing these possibilities is time. When the lifetime of an ion pair exceeds a few picoseconds, as is generally true for highly stabilized carbocations, then its internal energy and solvation will reach equilibrium. The conventional Winstein formulation may then apply fully. When the carbocations have shorter lifetimes, the physical reality is just more complex.

We would suggest that this expanded description of ion pairs accounts for many observations in the literature, observations that otherwise appear shoehorned into mechanistic models not considering dynamic effects. We briefly describe here a series of examples. This is unavoidably speculative, but suggestive of experimental and computational studies that could assess the role of solvation dynamics in fundamental aspects of carbocation reactions.

A historically notable example is the addition of DCl to norbornene (11). This reaction does not give an equal mixture of nonrearranged product 12 and rearranged product 13,⁴² despite extensive evidence that the intermediate “2-norbornyl” cation itself is nonclassical with C_s symmetry. Nordlander had attempted to explain this observation based on the equilibrating ion-pair ideas that he had used to account for observations with 1, but the structure and nature of the requisite unsymmetrical but equilibrating ion pair intermediates was nebulous. The experimental observations in fact merely require that the initial positioning of the chloride ion and surrounding solvent molecules be unsymmetrical, as will necessarily follow from the protonation transition state, and that a portion of the initial ion pairs react rapidly before achieving equilibrium solvation. As we have recently shown in another context,⁴³ when product selectivity is decided without barrier, initial asymmetry in structure readily leads to unequal amounts of otherwise equivalent products.

A broader if less notorious historical controversy was regarding the origin of S_N1 rate accelerations in nucleophilic solvents and the role of nucleophilic solvation (né nucleophilic solvent participation⁴⁴) versus reversible ionization and “hidden return” of ion pairs.⁴⁵ A preponderance of data has favored the importance of nucleophilic solvation, at least for secondary substrates, but aspects of the complex debate long persisted.⁴⁶ An early lynchpin observation by Shiner, appearing to strongly favor the importance of hidden return, was that the addition of 4-bromobenzenesulfonic acid (14) to propene in trifluoroacetic acid affords the isopropyl brosylate 15 instead of the isopropyl trifluoroacetate 16 adduct from solvent.⁴⁷ Shiner argued that since the tight ion pair arising from the protonation predominantly collapses to 15 instead of reacting with solvent, then the S_N1 reaction of 15 must involve a rate-limiting step that is later than formation of the tight ion pair, and so at the formation of a solvent-separated ion pair. This idea influenced the interpretation of a substantial history of observations.⁴⁸ The flaw in this argument, as suggested by the present work, is the assumption that there is only one tight ion pair operative in the system. We would suggest that a nonequilibrium solvated ion pair is initially formed from protonation of propene by 14, and that this ion pair rapidly collapses to 15. This then undercuts the original support for the importance of solvent-separated ion pairs in the solvolysis of 15 and other secondary systems.⁴⁸

We would note that in both of the above controversies, the ultimately disfavored side was supported by observations that could not have been comfortably interpreted otherwise employing the ideas available at the time. The expansion of ideas to include a role for solvation dynamics then allows explanations that do not conflict with the prevailing view.

Undergraduate organic chemistry textbooks routinely describe the rearrangements of secondary carbocations to tertiary cations that accompany the additions of acids to alkenes.⁴⁹ A notable aspect of these rearrangements, exemplified by the addition of HCl to tert-butylethylene (17)⁵⁰, is that mixtures of nonrearranged and rearranged products (e.g., 18 and 19) are often produced. This becomes surprising when it is recognized that there is no calculated barrier for the highly exothermic conversion of 20 to 21,⁵¹ as is typical for secondary to tertiary rearrangements. The cation 20 can only exist dynamically for a short time, with simulations placing the median lifetime at around 300 fs.^51b If these reactions involved only solvent-equilibrated ion pairs with lifetimes crudely on the order of that seen for solvent-equilibrated ion pairs here, then significant amounts of unrearranged products could not be formed. We would suggest that these reactions may initially form nonequilibrium solvated ion pairs that react rapidly to mainly afford unrearranged products, then mainly afford rearranged products as solvent-equilibrated ion pairs arise and the ion-pair lifetimes increase. The mixture of products would then be decided dynamically on a bifurcating surface rather than representing the crossing of competing barriers.

S_N1 reactions of RX have often exhibited a striking memory effect when the nature of their ion pairs has been probed. That is, studies of ¹⁸O or ¹⁷O isotopic equilibration and racemization or allylic transposition in recovered reactants have evidenced a definable structure in the ion pairs. The detailed observation varies: sometimes allylic transposition or racemization occurs faster than isotopic equilibration,^19a,d,52,53 sometimes the reverse is true,^19c sometimes the transposition is accompanied by bond formation to a new oxygen,^19a,d sometimes preferentially to the original oxygen.^53,54. A prominent example occurs in the solvolysis of 2-norbornyl brosylate, where racemization is faster than ¹⁷O exchange.⁵³ Another example is the solvolysis of 22, in which racemization from formation of 23 is faster than ¹⁸O equilibration by formation of 24. This suggests that the ion pairs are very short-lived. In contrast, the products in these reactions, as in 2-norbornyl solvolyses or the formation of 25, have generally lost all memory of the original structure.^19,52,53 A possible economical explanation for the contrasting features of these reactions is that the return to isomerized reactant involves a short-lived nonequilibrium solvated ion pair, while product formation involves longer-lived equilibrium ion pairs. As is consistent with this idea, changes in solvent polarity (either by changing the solvent or by adding salts) have a larger effect on the rate of product formation than on isomerization or isotopic equilibration.^19a,c,d

The era of carbocation chemistry in the twentieth century featured some of the most elegant experiments in chemical history, experiments that defined subtle features of these very complex mechanisms but that occasionally led to conflicting interpretations. We view it as probable that some misinterpretations and conflicts arose from the assumed foundational idea that a reaction mechanism in solution is fully defined by its sequence of intermediates and transition states. This then led to the implicit expectation that identical solute structures should behave identically, and the corollary that differing behavior implied differing intermediates. The nature of an intermediate and its subsequent reactions will however depend on its solvation and its dynamics, which in turn depends of how the intermediate was prepared⁵⁵ and its lifetime. The results here show that this can have a substantial effect on observations for short-lived cations, suggesting the need for some modern reconsideration of the results from the era.