Norcaradiene–Cycloheptatriene Equilibrium: A Heavy-Atom Quantum Tunneling Case

The equilibrium between norcaradiene and cycloheptatriene, which has captivated chemists for more than half a century, is revisited by state-of-the-art quantum chemical calculations. Our theoretical data significantly deviate from the experimental results (J. Am. Chem. Soc.,1981,26, 7791–7792), especially at low temperatures, where isomerization is dominated by heavy-atom tunneling. This effect results in an extremely short half-life for norcaradiene, rendering it undetectable. This work sheds light on this equilibrium, updating the kinetic and thermodynamic data while also expanding the repertoire of organic reactions controlled by this exotic quantum effect.


Electronic Structure Calculations
Norcaradiene, cycloheptatriene and the saddle point linking the two isomers were optimized with the double hybrid revDSD-PBEP86 1,2 in combination with the D3BJ empirical dispersion correction 3,4 and the correlation consistent basis set jun-cc-pVTZ 5,6,7 in vacuum.Hessian evaluation was carried out at the same level of theory finding none and one imaginary frequencies for energy minima and saddle point, respectively.Energy refinements were carried out by extrapolating to the complete basis set limit with a mixed Gaussian/exponential expression for cardinal numbers n = 3, 4 and 5 solving the following system of equations: 8,9,10 {  ( 1 ) =   +  −( 1 −1) +  −( 1 −1) 2  ( 2 ) =   +  −( 2 −1) +  −( 2 −1) 2  ( 3 ) =   +  −( 3 −1) +  −( 3 −1) 2 To validate the level of theory, we also corrected the electronic energies with the CCSD(T)-F12 11.12 on the revDSD-PBEP86(D3BJ)/jun-cc-pVTZ geometries (abbreviated as revDSD).As auxiliary and special orbital basis set, the cc-pVTZ-F12-CABS and cc-pVTZ-F12 were used.Comparative results are gathered in Table S2.As inferred from Table S2 the revDSD-PBEP86(D3BJ)/CBS energies give very similar results to those calculated with Coupled Cluster.The result exhibiting the largest deviation from the CCSD(T)-F12 energies is the relative stability between the isomers (0.39 kcal/mol).This small difference is not critical for the conclusions attained in this work that focuses mainly on the kinetics of the isomerization from norcaradiene to cyclocheptatriene.The T1 diagnostic reveals that static correlation calculations are unnecessary for this system.Accordingly, our reference level of theory through this work is revDSD-PBEP86(D3BJ)/CBS for the calculation of energies on revDSD-PBEP86(D3BJ)/jun-cc-pVTZ geometries.
Since high-frequency vibrational modes in the transition state are important for predicting quantitative rate constants at low temperatures, anharmonic zero-point vibrational energies corrections (ZPEAnh) were evaluated at revDSD-PBEP86(D3BJ)/jun-cc-pVTZ within vibrational perturbative theory to second order (VPT2). 13For calculating the scaling factor to account for high frequency anharmonicity, the following expression was employed: 14  ℎ = where ωAnh and ωHar are the anharmonic and harmonic ZPE energies computed at the abovementioned level of theory, respectively.λ ZPE is the scaling factor used to correct harmonicities.The results obtained are shown in Table S3.

Kinetic Calculations
In order to reproduce the experimental results conducted by Rubin, 15 our kinetic calculations were performed taking solvent effects into consideration.Thus, all calculations conducted up to this point have been repeated using the continuum solvation model (SMD) 16,17 in cyclohexane.
For kinetic calculations, we used the canonical variational transition state theory (CVT) 18 employing the small curvature tunneling approximation (SCT) 18 across the temperature (T) range of 50-500 K, and using the Pilgrim software. 19The expression for unimolecular rate constants is provided by: where R represents the ideal gas constant, κ SCT is the small curvature multidimensional tunneling transmission coefficient, and Г CVT is the canonical variational transition state recrossing coefficient.This coefficient is defined as k CVT /k TST , where k TST is the rate coefficient of conventional transition state theory, and k CVT is the rate coefficient of canonical variational transition state theory.Additionally, KB stands for the Boltzmann constant, and h denotes the Planck constant.
Q VT and Q R are the total partition functions of the variational transition state and the reactant, respectively, while E VT represents the potential energy of the variational transition state.
The energies along the minimum energy path (MEP), obtained using the revDSD-PBEP86(D3BJ)/jun-cc-pVTZ method in cyclohexane (SMD), were corrected with the revDSD-PBEP86(D3BJ)/CBS in cyclohexane (SMD), employing the interpolated single-point energies (ISPE) algorithm.For correcting the MEP we used the two isomers, the saddle point and three points in the backward direction (S<0), as well as three additional points in the forward direction (S>0).
To account for the high frequency anharmonicities, we multiplied the harmonic frequencies of the norcaradiene and cycloheptatriene by the scaling factor 0.988.The same scaling factor was used for the saddle point and the whole MEP.All the electronic and kinetic calculations for the isomerization between 14 and 15 have been carried out with the same methodology as detailed in sections 2.1 and 2.2, but changing the solvent to argon (SMD).Arrhenius fitting for the reverse isomerization proposed by Rubin, and based on data reported by Huisgen for compound 13. 15,20

Breakdown of Notes a and b in Table
The assumption reached by Huisgen to estimate the population between norcaradiene and cycloheptatriene is based on bicyclo[4.2.0]octa-2,4-diene, as norcaradiene could not be detected.He stated the following verbatim: The Diels-Alder adducts of cycloheptatriene (C) are structurally derived from norcaradiene (D).Kinetic measurements on the reaction with tetracyanoethylene gave kd values that were proportional to the dienophile concentration over the entire range measured.This means that the establishment of the equilibrium (C)+(D) is fast in comparison with the Diels-Alder reaction.The intermediate cannot be detected kinetically in this case.If it is assumed that the diene activity of (D) can be equated to that of (B) (equal k2 values for the addition of tetracyanoethylene to the cyclohexadiene portion of the two molecules), a 0.1 % equilibrium concentration of (D) at 20 "C would lead to kd values of the order found.
Then, based on the above-mentioned assumption, Rubin stated: It has been suggested (quoting Huisgen) that the equilibrium concentration of 1 (referred to norcaradiene) in 2 (referred to cycloheptatriene) at 20 °C is 0.1%.Using the equilibrium constant based on this proposal, the free energy difference between 1 and 2 at 25 °C is of the order of -4 kcal/mol and  1→2 ‡ ≈11 kcal/mol.Assuming that the entropy of activation for 2 → 1 is close to zero, the approximate Arrhenius expression becomes
revDSD energies in cyclohexane (SMD) for the stationary points and the six points along the minimum energy path used for the correction of the energy profile for the isomerization between 1 and 2. f (forward) and b (backward) indicate the sign of the reaction coordinate.ZPE are given in kcal

Table S4 . Calculated and Experimental Rate Constants in s -1 for the Isomerization Between Norcaradiene and Cycloheptatriene, Calculated Free Energy of Reaction in kcal/mol and Equilibrium Constants
Inter-and extrapolation of the Rubin's Arrhenius Fitting obtained with his three measurements at 93, 98, and 103 K. 15 b a