Supramolecular Catalysis of m-Xylene Isomerization by Cucurbiturils: Transition State Stabilization, Vibrational Coupling, and Dynamic Binding Equilibrium

The ability of cucurbit[6]uril (CB6) and cucurbit[7]uril (CB7) to catalyze the thermally activated 1,2-methyl shift isomerization pathway of m-xylene in vacuum is investigated using infrequent metadynamics. CB6 is predicted to effectively and selectively catalyze the meta-to-para isomerization through stabilization of the transition state (TS) by van der Waals push (packing coefficient ≈74%), while inhibiting the meta-to-ortho pathway by molding effects of the cavity. Interestingly, despite the snug binding, a very low rate of host–guest vibrational energy transfer is revealed using a novel approach of host–guest partition of the mode-specific anharmonic relaxation rates and ab initio molecular dynamics. The weak vibrational coupling suggests that CB can act as a thermal buffer, possibly shielding encapsulated guests from outside vibrational perturbations such as solvent effects. This dynamic effect could provide an additional boost to the reaction rate by blocking the occurrence of reaction barrier recrossing caused by the friction with surrounding molecules. Finally, mean residence times of xylene into the hosts’ cavity were estimated for a range of host–guest complexes, revealing a highly dynamic equilibrium allowing very high guest turnover rates that could minimize catalyst inhibition effects commonly suffered by other supramolecular catalysts.

S7 Assignment of the complex modes to the guest as X (orange ), the host as C (blue ×) or mixed as M (green ) for the [m-xylene+H] + @CB6 system. . S14 S8 Assignment of the complex modes to the guest as X (orange ), the host as C (blue ×) or mixed as M (green ) for the m-xylene@CB7 system. . . . . S15 S9 Assignment of the complex modes to the guest as X (orange ), the host as C (blue ×) or mixed as M (green ) for the [m-xylene+H] + @CB7 system. . S15 S10 Eyring plots obtained using Eq. 3 in the case of cavity exiting. . . . . . . . . S16 List of Tables S1 Mean times to reaction at 300 K (t 300K react ), 400 K (t 400K react ), 500 K (t 500K react ), 600 K

S4
The biased trajectory was allowed to run for 100 ps after equilibration with Gaussians added every 500 timesteps with a height of 0.5 kcal/mol and a width of 0.05 for both C 1 and C 2 .
The collective variable C 1 is defined as the coordination number of carbon 1 to carbon 2 while C 2 is defined as the coordination number of carbon 1 to carbon 4 in the case of the MP ‡ TS and carbon 1 to carbon 3 in the case of the MO ‡ TS. The collective variables are defined as: with nn=8, nd=14 and R 0 =1.8Å. 6 All trajectories were computed using CP2K 7 with a timestep of 1 fs using the semiempirical PM6-D3 model. 2,3 The target accuracy of the SCF cycle procedure was set to 5×10 −7 Ha and the ASPC extrapolation method was used to help with energy conservation. 8 The time to reaction t MTD is defined as the time needed for C 1 to become lower or equal to 0.3 and C 2 to become higher or equal to 0.7.

Host-guest vibronic coupling by using third order anharmonic decay rates
The relaxation rate of the k th mode γ k for the m-xylene@CB6, [m-xylene+H] + @CB7 systems are, respectively, 372, 375, 426 and 429. Since it is assumed that frequencies will not be significantly shifted by the relaxation rate, the system is solved S5 using the frequencies obtained from the normal mode analysis using the harmonic approximation. Initial guesses for the γ k was 150 cm −1 and iterated until the sum of the residue became smaller than 1×10 −4 cm −1 .
The anharmonic cubic coefficients X ijk (units given in cm −1 ) were computed using Gaus-sian16 on geometries optimized to the very tight level with an ultrafine integration grid. The Boltzmann's population of the i th mode, with frequency ω i , is represented by n i . Vibrational modes of host-guest complexes are formally distributed over all atoms, host and guest alike, that form the complex. In practice however, some vibrational modes can be predominantly located on the guest (host) with the amplitude of the associated eigenvector on the atoms of the guest (host) being high while being low or negligible on the atoms of the host (guest). To quantify the degree to which a mode m is located on the guest, a parameter S m is defined.
To compute S m , the eigenvector associated with mode m, written a m , is first normed and sorted so that its first 3N components, corresponding to the displacements of the guest's N atoms, appear first. The vector a m is then truncated to these 3N first components to yield a m and the norm of this new vector is computed to yield S m . A value of 1 for S m reveals that mode m is entirely located on the guest while a value of 0 means that mode m is entirely located on the host. Eventually, one can write S m as:

Host-guest vibronic coupling by molecular dynamics
Computation of the efficiency of the kinetic energy transfer between the CB6 host and a A molecular simulation is then started from the equilibrium configuration where the initial speeds v n,x , v n,y and v n,z of the n atoms is given by the eigenvector associated with the 368 th mode appropriately scaled so that the total energy E pot written as: velocities v n,x , v n,y and v n,z as follows: 10 where the index i loops over the variable x, y, z. Power spectra for parts of the trajectory were obtained by shifting t end and the origin of time to the desired time region. The geometry optimization, vibrational analysis and trajectory computation were all carried out using CP2K at the PM6-D3 level using the ASPC extrapolation method.
Entropy values for [m-xylene+H] + , CB6 and [m-xylene+H] + @CB6 were obtained by optimising each structure then computing their respective normal modes at the PM6D3 level of theory using Gaussian 16.

Metadynamics of host-guest dissociation kinetics
The estimation of the cavity escape rates from metadynamics was performed for each of   Figure S4: Illustration of the breakdown of γ k into its contributions γ xx k , γ xc k , γ xm k , γ cc k , γ mm k , γ cm k in the special case where the k th mode belongs to the guest. The X part of the diagram (X standing for xylene) shows the modes assigned to the guest (S≥0.9), while the C part of the diagram (C standing for cucurbituril) contains the modes assigned to the host (S≤0.1).
Eventually the M part of the diagram (M standing for mixed) contains the modes that could not be unambiguously assigned to either the host or the guest. 3 Tables   Table S1: Mean times to reaction at 300 K (t 300K react ), 400 K (t 400K react ), 500 K (t 500K react ), 600 K (t 600K react ) and 700 K (t 700K react ). Table S2: Main contributions to the relaxation rate of mode k=368. The numbers i and j represent the numbers of the modes involved in the relaxation pathway with the xylene localization number given between parenthesis.  Table S3: Main contributions to the relaxation rate of mode k=318. The numbers i and j represent the numbers of the modes involved in the relaxation pathway with the xylene localization number given between parenthesis.  Table S4: Main contributions to the relaxation rate of mode k=303. The numbers i and j represent the numbers of the modes involved in the relaxation pathway with the xylene localization number given between parenthesis.  Table S5: Main contributions to the relaxation rate of mode k=2. The numbers i and j represent the numbers of the modes involved in the relaxation pathway with the xylene localization number given between parenthesis.  Table S6: Main contributions to the relaxation rate of mode k=1. The numbers i and j represent the numbers of the modes involved in the relaxation pathway with the xylene localization number given between parenthesis.

Videos
The movies correspond to trajectories at 300 K obtained with a timestep of 1 fs and a CSVR