Influence of Equatorial CH⋅⋅⋅O Interactions on Secondary Kinetic Isotope Effects for Methyl Transfer

Abstract DFT calculations for methyl cation complexed within a constrained cage of water molecules permit the controlled manipulation of the “axial” donor/acceptor distance and the “equatorial” distance to hydrogen‐bond acceptors. The kinetic isotope effect k(CH3)/k(CT3) for methyl transfer within a cage with a short axial distance becomes less inverse for shorter equatorial C⋅⋅⋅O distances: a decrease of 0.5 Å results in a 3 % increase at 298 K. Kinetic isotope effects in AdoMet‐dependent methyltransferases may be m∧odulated by CH⋅⋅⋅O hydrogen bonding, and factors other than axial compression may contribute, at least partially, to recently reported isotope‐effect variations for catechol‐O‐methyltransferase and its mutant structures.

. Z-matrix template for TS cage complexes 5 Table S2. Optimised parameters for cage transition structures 6 Note on Table S2  6   Table S3. Z-matrix template for RS cage complexes 7 Table S4. Optimised parameters for cage reactant structures 8 Notes on Table S4  8   Table S5. B3LYP/aug-cc-pVDZ optimised energies for RS and TS structures (total energies in hartree, relative energies in kJ mol -1 ). 9 Notes on Table S5  9   Table S6. Vibrational wavenumbers (cm -1 ) for symmetric and antisymmetric (italics) axial

C-O bond stretching in transition structures of methyl cation in constrained water cages 10
Notes on Table S6  10   Table S7. Imaginary vibrational wavenumbers (cm -1 ) for motions of constrained cage water molecules. Vibrations in brackets correspond to the heavier isotopologue, vibrations in regular type to the TS, and in bold to the RS 11 Notes on Table S7  14   Table S8. B3LYP/aug-cc-pVDZ average CH-bond stretching force constants (aJ Å -2 ) for RS and TS.
Notes on Table S8  15   Table S9. B3LYP/aug-cc-pVDZ isotopic partition function ratios, 2° -D3 isotope effects for transfer of methyl cation from vacuum to the center of the constrained water cage with req = 3.0 Å at 298 K and isotope effect factors. 16 Notes on Table S9  16   Table S10. B3LYP/aug-cc-pVDZ isotopic partition function ratios, 2° -D3 isotope effects for insertion of symmetric axial structures [H2O…CH3 + …OH2] from vacuum to the center of the constrained three-water equatorial cage at 298 K and isotope effect factors. 17 Notes on Table S10  17   Table S11. B3LYP/aug-cc-pVDZ isotopic partition function ratios and factors for RS and TS, and KIEs, for methyl transfer within a constrained superheavy cage with rax = 2.04 Å. 18 Notes on Table S11. 19

Computational Methods
Constrained geometry optimizations and analytical second-derivative calculations of Hessians were performed by means of the Gaussian09 program (revision A.02). 1 The B3LYP density functional was used with the aug-cc-PVDZ basis set, a method previously identified as performing well for a harmonic vibrational frequencies, leading to reliable estimates of isotope effects. 2 Residual translational and rotational contributions to Hessian elements (punch=derivatives) were removed by a projection method 3 and equilibrium and/or kinetic isotope effects at 298.15 K were determined within the rigid-rotor/harmonic approximation without scaling of vibrational frequencies, which satisfied the Teller-Redlich product rule; tunneling was treated by means of Bell's model applied to the imaginary frequency for motion along the reaction coordinate. 4 The conventional transition-state theory treatment of KIEs with the Born-Oppenheimer, rigid-rotor and harmonic oscillator approximations (and neglecting tunnelling and transmission-factor contributions) considers the rate-constant ratio k/k' (where the prime denotes the rate constant for the heavier isotopologue) as the product of three factors (eq. 1) involving translational, rotational and vibrational partition functions for the isotopologous RS and TS species. 4,5 The first is the mass and moments-of-inertia factor (MMI), where | I | is the determinant of the moment-of-inertia tensor (i.e. the product of the three principal moments of inertia) of a non-linear N-atomic molecule of molecular mass M; rotational symmetry factors are omitted here since they are all equal to unity for the applications to be presented below. The second is the Boltzmann excitational (EXC) factor corresponding to the relative populations of the higher vibrational quantum states, and the third is the zero-point energy (ZPE). EXC involves a product, and ZPE a summation, over 3N -6 and 3N -7 real vibrational frequencies of RS and TS, respectively, where u = hc/kBT with frequency (s -1 ) replaced by wavenumber  (cm -1 ) and h, kB, c and T are the Planck and Boltzmann constants, the velocity of light and the absolute temperature. Inspection of the forms of the MMI, EXC and ZPE factors shows that the KIE may be rewritten simply as a quotient of isotopic partition function ratios (IPFRs) for RS and TS (eq. 2).
(1) (2) It is common to replace MMI in eq. 1 by an equivalent factor containing only vibrational frequencies. According to the Teller-Redlich product rule, 6 the masses and moments of inertia for a pair of isotopologues are related to the vibrational frequencies: the equality expressed by eq. 3 assumes separability of translational and rotational motions from vibrational motions within the harmonic approximation. Substituting the vibrational product VP for each of RS and for TS in eq. 1, the KIE may be written as eq. 4, the Bigeleisen equation, where VPR is the vibrational product ratio. 7,8 Two points should be noted: (i) the product of ratios of atomic masses m that appears eq. 3 vanishes from the KIE because it is identical for both RS and TS; (ii) it is conventional to consider the ratio of imaginary transition frequencies for the TS as a separate factor, so that (in eq. 5) the VPR and EXC products and the ZPE summation are all taken over 3N -7 real TS frequencies. (3) Use of the Bigeleisen equation enables the determination of KIEs from normal-mode frequencies without the need to consider the MMI factor explicitly. However, there is practical merit in separately determining the vibrational product and mass-moment-of-inertia terms for a pair of isotopologues, rather than relying upon their equality: the requirement for VP and MI to be equal in value provides a strict test for the correctness of the vibrational frequency calculations. If these two terms are not equal to a satisfactory number of decimal places, it is a sure indication that something is wrong, often (in our experience) with the definition of internal coordinates used in the projection method for removing residual components of translational and rotational motion from the Hessian.
We prefer to evaluate IPFRs for each RS or TS species separately and then to evaluate kinetic isotope effects (KIEs) by means of eq. 2 or equilibrium isotope effects (EIEs) by the analogous eq. 6, where PS refers to a minimum-energy product structure.
Within the inherent assumption of separability in this treatment of KIEs, the quantum correction  ‡/ ‡' for motion along the transition vector may be approximated by Bell's expression 9 for an inverted parabola, eq. 7; this quantum correction is applied routinely to a transition frequency in evaluation of the IPFR for isotopologous TSs.      Table S5: E ‡ is the barrier height for methyl transfer between axial waters within the constrained cage, either with the three equatorial waters present (req = 3.0, 3.5, 3.75 and 4.0 Å) or without them (req = ).
On the one hand, the value of this barrier increases steeply as the methyl-transfer distance ( = 2  rax ) increases; this may be rationalized simply in terms of the Principle of Least Nuclear Motion, as has been noted previously. 11 On the other hand, the value of this barrier decreases as the CH…O distance (req) decreases. This reflects a differential between the RS and TS energies for interaction between the each of the symmetric axial structures [H2O…CH3 + …OH2] and the three equatorial water molecules which serves to stabilize the TS more effectively for shorter, stronger CH…O hydrogen-bonding interactions. (These interactions give rise to catalysis within the constrained cage model.)  Table S6: The antisymmetric stretching mode is the reaction-coordinate mode (or transition vector) for methyl transfer within the cage. Its frequency is imaginary (as expected) in all cases. Replacement of protium in CH3 by deuterium in CD3 leads to a diminution in magnitude of this frequency (as expected). An increase either in the value of req for a particular value of rax or in the value of rax for a particular value of req leads to an increase in the magnitude of this frequency, consistent with an increase in the barrier height E ‡ for methyl transfer (see Table S5).
The symmetric stretching frequency is insensitive to isotopic substitution in the methyl group because atomic motions of these atoms do not contribute to this normal mode of vibration. This frequency has a real value for rax = 2.04 Å but is imaginary for rax = 2.525 and 3.0 Å. However, it is important to recognise that these symmetric structures do not correspond to stationary points on the potential energy surface for the full unconstrained system: in particular, the gradient of the energy with respect to displacement in the symmetric stretching coordinate is not zero. In the "superheavy" cage, all frequencies are lowered and the isotopically-sensitive modes of the methyl group are decoupled from those of the cage environment.

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Continued on next page…  Table S7.
Owing to the fact (mentioned above in the notes on Table S6) that neither the RS nor the TS structures correspond to stationary points on the potential energy surface for the full unconstrained system, and also because of the imposed symmetry, there are a number of additional modes associated with imaginary vibrational frequencies. Four of these are evident in the req =  structures without the three equatorial waters, and involve in-phase and out-of-phase combinations of wagging and twisting of the axial water molecules. It is obvious that the CH3OH2 + fragment of the constrained RS structures is unstable with respect to displacement towards a pyramidalized sp 3 O atom instead of the trigonal-planar geometry imposed by the constraints, an this instability is manifest in both the RS and TS cage structures. The constrained structures represent saddle points for inversion and torsion of the axial OH2 moieties. However, in practice these modes are almost insensitive to isotopic substitution in the methyl group and therefore do not affect the calculated KIEs.
In the cage structures with req = 3.0, 3.5, 3.75 and 4.0 Å, some of the modes involving the axial waters are strongly coupled with motions of the equatorial waters and the brief descriptions given in the Table are  The quantities MI and VP represent the "masses & moments of inertia" and "vibrational product" terms appearing in eq.3. According to the Teller-Redlich Product Rule, 6 these terms should be equal in value. The extent to which this is found to be true in practice provides a stern test of the correctness and internal consistency of the vibrational frequency calculations, and especially of the method employed to project out contaminating components of translation and rotation from the vibrational degrees of freedom. Inspection of the results presented in Table S7 shows agreement in all cases to the fourth decimal place, with the majority being to the seventh.
The normal MMI factor is almost constant. For each value of rax the EXC becomes less inverse as req decreases but the ZPE factor becomes more inverse and dominates the trend in the EIE. Some of these data are presented graphically in Figure 2 of the paper.
Hydrogen atoms on "water" molecules have mass mH = 999; hydrogen atoms on methyl group have mH = 1.
Quantum corrections on transition frequencies are included in the IPFRs for TSs. The quantum corrections (QC) on the KIEs are all very close to unity, indicating the relative insignificance of tunneling in these model methyl transfer reactions. Some of these data are presented graphically in Figure 3 of the paper.