Virtual transition states

Reactions in parallel

A trivial example of two reaction pathways in parallel is where achiral reactants lead to an achiral product by means of a pair of enantiomeric TSs. Consider the addition of a single water molecule to formaldehyde (in a gaseous reaction) yielding methanediol, H₂O + CH₂=O → HOCH₂OH. As it happens, this was the first published example of a transition structure computed by means of ab initio gradient methods, ⁶ but the significant point is that the transition structure is chiral (Fig. 2). If observed rate constants for this reaction with (say) H₂O and D₂O were to be compared, the KIE would inevitably be interpreted as arising from a symmetric TS structure (in which the non-transferring hydrogen atom would be coplanar with the heavy atoms and the transferring hydrogen) whereas in fact it would arise from equal contributions of enantiomeric TSs: the inferred structure would be that of the virtual TS.

Fig. 2:

Computed transition structure (HF/4-31G) for addition of water to formaldehyde: (a) ‘side’ view, (b) ‘top’ view.

A less trivial example of parallel TSs for competing reaction pathways comes from consideration of 4,4′-dimethoxybenzhydrylpyridinium cation solvolysis in ethanol. There are four reactant conformers that differ only in the relative orientations of the methoxy substituents, and each is linked directly to a specific transition structure for heterolysis of the scissile C-N bond leading to formation of a benzhydryl cation intermediate that then undergoes nucleophilic attack by ethanol (Scheme 1). It was demonstrated recently that the correct apparent Gibbs energy of activation for N multiple reaction pathways in parallel, arising from a set of rapidly interconverting reactant conformers (i.e. Curtin-Hammett conditions ⁷ ), may be obtained by eqn (1), which derives the fact that the observed rate coefficient is the sum of the individual rate constants, one for each separate pathway. This equation involves the Gibbs energy Δ^‡ G _i for each individual TS conformer i relative to the reactant-structure conformer of lowest Gibbs energy, ⁸ where x ₀ is the equilibrium mole fraction of that lowest-energy reactant conformer. The term within the curly brackets is equivalent to the ratio of partition functions for the TS and reactant state (RS), but it is convenient to evaluate it using computed values of Gibbs energies for the relevant structures.

Scheme 1:

Top: heterolysis of one conformer of 4,4′-dimethoxybenzhydrylpyridinium cation. Bottom: cartoon representations for other reactant conformers a, b, c and d to illustrate methoxy-group orientations.

Figure 3 shows Gibbs energies computed ⁸ for heterolysis of these cations at the M06-2X/6-311+G(2d,p)/PCM=EtOH level of density functional theory (298 K, 1 M), and all are relative to the energy of conformer a; the respective mole fractions are obtained as normalised Boltzmann populations (at 298.15 K): x ₀ = x _a = 0.52, x _b = 0.15, x _c = 0.17, x _d = 0.15. All four reactant conformers lie within 3 kJ mol⁻¹ of each other and are interconverted by internal rotations involving transition structures (dashed black lines) for clockwise directions of rotation. There are separate transition structures for anticlockwise rotations, but their relative energies all lie within the range shown in Fig. 3 (17.0–19.5 kJ mol⁻¹), which are very low as compared with the individual activation Gibbs energies for heterolysis (indicated by vertical blue lines), and the system therefore satisfies Curtin-Hammett conditions. Application of eqn (1) yields an apparent Gibbs energy of activation ∆ ‡ G app parallel for heterolysis equal to 91.6 kJ mol⁻¹, a value lower than any of the individual values of Δ^‡ G _i . This is a characteristic feature of reactions occurring by multiple pathways in parallel. The largest share of the total reaction flux between reactants and products proceeds via the transition structure with the lowest Gibbs energy, and the other transition structures contribute smaller shares in proportion to their Boltzmann populations. It is clear that each value of Δ^‡ G _i corresponds to an individual transition structure, which approximately represents the TS for a separate reaction pathway, but to what does the apparent Gibbs energy of activation ∆ ‡ G app parallel correspond? The answer, of course, is that it corresponds to the virtual TS for the overall reaction: it is a weighted average of all four individual contributing TSs, and it is not a real physical entity. To quote Stein: ⁵ ‘For reactions in which more than a single TS is kinetically significant, any experiment designed to probe the structure of the rate-limiting TS will yield information not about a single, real TS, but rather about a “virtual” TS whose structure is a weighted average of structures of the several rate-limiting, real TSs.’

Fig. 3:

Computed Gibbs energies [M06-2X/6-311+G(2d,p)/PCM=EtOH] for heterolysis of 4,4′-dimethoxybenzhydrylpyridinium cation (298 K, 1 M), all relative to reactant conformer a. Energies of individual TSs for heterolysis are depicted by vertical blue lines, and for interconversion of reactant conformers are depicted by dashed back lines.

It is well known that a Hammett plot for a reaction of substituted-aryl substrates involving a change in mechanism displays concave-upward behaviour, an example of which is the electrophilic bromination of trans-monosubstituted stilbenes in methanol (Scheme 2). ⁹ The logarithms of reduced second-order rate constants {k _obs} for the most strongly electron-donating substituents correlate with σ ⁺ giving ρ = −5.56 (red points, Fig. 4), while electron-withdrawing substituents yield ρ = −1.65 when correlated against σ (green points, Fig. 4). (Note: {k _obs} = k _obs/[k _obs], where [k _obs] represents the units of k _obs, ¹⁰ in this case M⁻¹ min⁻¹; much common usage and many textbooks incorrectly attempt to construct Hammett correlations using dimensioned rate constants, whereas a logarithm only has strict mathematical meaning for a quantity with dimension = 1.) The solid black curve in Fig. 4 is obtained from the logarithm of the apparent rate constant given by eqn (2). The blue points represent the observed rate constants for X = m-Me and H: they lie above both the extrapolated red and green dotted lines but are correlated well by the solid black curve, demonstrating that both competing mechanisms contribute to the apparent rate constant, namely parallel pathways via TSs resembling either an open carbocation (strongly electron-donating substituents, Scheme 2) or a cyclic bromonium cation (electron withdrawing substituents, Scheme 2). The virtual TS corresponding to this region of the Hammett plot is an average of these two real TSs.

With reference to the earlier discussion of trajectories across a FES (Fig. 1), each of the open carbocation and cyclic bromonium cation pathways (Scheme 2) corresponds to a distinct swathe of microscopic trajectories around a particular minimum free-energy path. The overall FES contains two separate saddle regions separated by a local maximum, and there is a unique FES for each substituent group X with different weightings for the pair of real TSs in parallel. The virtual TS should not be confused with the averaging of microscopic trajectories for a single reaction pathway.

Scheme 2:

Alternative mechanisms for electrophilic bromination of trans-monosubstituted stilbenes. Strongly electron-donating substituents favour the upper path via the open carbocation (red); electron withdrawing substituents favour the lower path via the cyclic bromonium cation (green).

Fig. 4:

Hammett plot for electrophilic bromination of stilbenes. Red points denote strongly electron substituents (vs. σ⁺), green points denote electron-withdrawing substituents (vs. σ), and blue points denote X = m-Me and H; the solid black curve is represented by eqn (2).

Reactions in series

A curious (although perhaps also trivial) example of a stepwise reaction involving multiple TSs in series arises from a computational study of the identity reaction H₂O +  ^t BuOH₂ ⁺ → ⁺H₂OBu ^t  + OH₂ at the HF and B3LYP/6-31G*/PCM=water levels of electronic structure theory, which occurs as a sequence of individual methyl-group rotations superimposed on an overall S_N2 Walden inversion; ¹¹ in practice this reaction would follow pseudo first-order kinetics. As expected, each TS is very carbocationic in character, with a ‘ ^t Bu⁺’ moiety sandwiched between a pair of water molecules. It may come as a surprise to recognise that, if this displacement reaction were to occur solely by Walden inversion coupled with asymmetric motion of the nucleophile and nucleofuge, the product would have its methyl groups eclipsing with the new C–O bond. Methyl-group rotation is therefore an essential component necessary to attain an identical staggered product; the process could occur in a single-step concerted reaction, but with the particular computational method employed in this study it is predicted to involve a series of three separate transition structures, as shown in Fig. 5. Inspection of the transition vector for each transition structure reveals a predominant rotation of just one methyl group at a time about the C–C bond connecting it to the carbocationic centre. The product complex possesses three staggered methyl groups (1 _sta , 2 _sta and 3 _sta ) like the reactant complex.

Fig. 5:

HF/6-31G*/PCM=water computed transition structures for the identity reaction H₂O +  ^t BuOH₂ ⁺ → ⁺H₂OBu ^t  + OH₂. (N.B. all optimised structures and the topography of the minimum-energy reaction path are qualitatively the same at the B3LYP/6-31G*/PCM=water level of computational theory.)

It should be noted that the use of a more accurate and reliable computational method than was available in 2001 ¹¹ might (or might not) yield a different picture – for example, a single transition structure with synchronous methyl-group rotations – but that is not the point of the present discussion, which is rather to illustrate the possibility that an experimentally derived transition-state structure may be a virtual TS.

The situation is actually more complicated than shown in Fig. 5 because each of the three steps can occur by either of two parallel paths, according to the direction of methyl-group rotation: details may be found in the original paper. ¹¹ For the present purpose it is sufficient to note that the transition structures for the first and third steps, and the two intermediates, are enantiomerically related, and the transition structure for the second step is also chiral. At the B3LYP/6-31G*/PCM=water level of calculation, the Pauling bond order of each C⋯O partial bond in the central transition structure is approximately 0.19, indicating a substantial degree of carbocationic character, but the transition vector has an imaginary frequency of 195i cm⁻¹, which is further evidence that the ^t Bu moiety does not lie in an energy minimum. The total electronic energy of the equivalent intermediates is 58 kJ mol⁻¹ higher than the reactant complex, with barriers of 6.8 and 11.1 kJ mol⁻¹ to the equivalent first/third and second transition structures, respectively: this implies that the intermediates would be very short-lived and thence that the steady-state approximation may be safely applied.

The apparent Gibbs energy of activation for a steady-state system of M reaction steps in series is given by eqn (3), where Δ^‡ G _j is the Gibbs energy of each sequential TS j relative to a common RS; this equation derives the fact that the reciprocal of the observed rate coefficient is the sum of the reciprocals of the individual rate constants, each relative to the same RS. Comparison with the similar-looking eqn (1) shows two important differences. First, since there is only one RS, its mole fraction is unity, and therefore x ₀ does not appear in eqn (3). Second, the latter equation has plus signs where eqn (1) has minus signs. The B3LYP/6-31G*/PCM=water values of Δ^‡ E _j for the three TSs in Fig. 5 are 64.8, 69.1 and 64.8 kJ mol⁻¹, respectively; assuming (for the purposes of illustration) that relative Gibbs energies may be approximated by relative total electronic energies, then insertion of these values into eqn (3) leads to an apparent activation energy of 69.9 kJ mol⁻¹, which is higher than any of the individual values of Δ^‡ E _j . This is a characteristic feature of reactions occurring by multiple steps in series: it is harder to cross the entire sequence of TSs in series than to surmount any single one. The extent to which each TS contributes to the overall apparent activation energy is given by eqn (4), where w _j is a weighting factor, otherwise known as the kinetic significance of TS j.

Using the three values of Δ^‡ E _j given above (and approximating Δ^‡ G by Δ^‡ E in eqn (4)), the respective weighting factors for the three TSs in Fig. 5 are 0.13, 0.74 and 0.13, which shows that the TS of most kinetic significance is that with the highest energy. These weighting factors are not mole fractions deriving from Boltzmann populations; perhaps they may be called anti-Boltzmann weightings. As with the example of parallel TSs considered above, the question to be asked is: to what does the apparent Gibbs energy of activation for a multi-step reaction with TSs in series correspond? And, again, the answer is a virtual TS. In the present example this may be considered as relating to nucleophilic displacement with Walden inversion, but with undefined methyl-group conformations; it is amusing to note that the virtual TS may be imagined as conforming to what an S_N2-like mechanism is expected to ‘look’ like despite each of the actual transition structures telling part of a somewhat different story.

Complex reactions in series and parallel

There are many complex reaction systems, including both enzyme-catalysed and nonenzymic examples, in which several different processes may simultaneously contribute to limiting the reaction rate. Consequently, any observed property (e.g. an isotope effect, or the slope of a linear free-energy relationship) that reports on TS-behaviour may be a complicated average of the effects on the elementary steps, reflecting a virtual TS. An early example, involving multiple steps in series and in parallel, considered variations in observed β-deuterium KIEs for hydrolysis of p-nitroacetanilide in basic solution. ¹² Not only did the observed KIE for p-NO₂C₆H₄NHCOCH₃ versus NO₂C₆H₄NHCOCD₃ change from 0.967 ± 0.011 in the least-basic solution, rise to about 0.98 with increasing hydroxide-ion concentration, and fall to 0.933 ± 0.020 in the most-basic solution, but also an apparently anomalous temperature dependence was found which would have required implausible values for the Arrhenius preexponential factor A _H/A _D and the isotopic activation energy difference ΔE _a if a single TS were responsible. The more reasonable explanation proposed was a shift in limitation of the rate away from the k ₁ TS (for formation of an anionic quasi-tetrahedral intermediate, Scheme 3) toward the k ₃ TS (for decomposition of a dianionic intermediate, Scheme 3), as [HO⁻] increased or as the temperature was raised. At low [HO⁻] decomposition of the adduct initially gives CL₃CO₂H + NO₂C₆H₄NH⁻ (where L = H or D), whereas at high [HO⁻] there is competing decomposition of the deprotonated adduct (initially to CL₃CO₂ ⁻ + NO₂C₆H₄NH⁻). The reciprocal of the phenomenological rate constant k ₀ (corrected from the observed rate constant for ionisation of the anilide substrate ¹² ) is given by eqn (5), where k ₂ = k′₂(k ₁/k ₋₁) and k ₃ = k′₃(k ₁/k ₋₁) are rate constants with respect to neutral substrate, and the β-deuterium KIE, k ₀(CH₃)/k ₀(CD₃), is given by eqn (6) with weighting factors as shown in eqn (7).

Scheme 3:

Stepwise mechanism for anilide hydrolysis in basic solution.