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A pictorial representation of zero point and tunnelling contributions to primary effects Rory More O’Ferrall

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Rory More O’Ferrall. A pictorial representation of zero point energy and tunnelling contributions to primary hydrogen isotope effects. Journal of Physical Organic , Wiley, 2010, 23 (7), pp.572. ￿10.1002/poc.1738￿. ￿hal-00552422￿

HAL Id: hal-00552422 https://hal.archives-ouvertes.fr/hal-00552422 Submitted on 6 Jan 2011

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Physical Organic Chemistry

A pictorial representation of zero point energy and tunnelling contributions to primary hydrogen isotope effects

For Peer Review

Journal: Journal of Physical Organic Chemistry

Manuscript ID: POC-09-0292.R2

Wiley - Manuscript type: Review Commentary

Date Submitted by the 12-Apr-2010 Author:

Complete List of Authors: More O'Ferrall, Rory; university college dublin, school of chemistry and chemical biology

Tunnelling , Westheimer effect, Hydrogen Isotope effects, Zero Keywords: point energy

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 31 Reaction coordinate and 'real' vibrations of the transition state 32 80x58mm (360 x 360 DPI) 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry Page 2 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Collinear collision complexes (a) early stage (b) at point of rebound 16 155x27mm (360 x 360 DPI) 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Page 3 of 32 Journal of Physical Organic Chemistry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Reaction coordinate diagram showing the vibrational of hydrogen within a collision complex 39 leading to tunnelling 91x80mm (360 x 360 DPI) 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry Page 4 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 surface showing minimum energy and tunnelling pathways 28 714x396mm (96 x 96 DPI) 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Page 5 of 32 Journal of Physical Organic Chemistry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 Calculated trajectory (dashed line) of collision complex subject to tunnelling 30 187x115mm (96 x 96 DPI) 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry Page 6 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Cross section along tunnelling coordinate cc' in Figure 3 showing overlap of zero point vibrational 33 functions 34 187x134mm (96 x 96 DPI) 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Page 7 of 32 Journal of Physical Organic Chemistry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 31 Projection of cuts across a potential surface onto a skewed reaction coordinate for hydrogen transfer 32 at the transition state and along a tunnelling coordinate 33 187x128mm (96 x 96 DPI) 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry Page 8 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 Classical barrier to reaction and a 'non-tunnelling 'reaction trajectory 31 153x98mm (360 x 360 DPI) 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Page 9 of 32 Journal of Physical Organic Chemistry

1 2 3 4 5 6 7 8 9 b 10 11 12 13 ZPE C 14 15 16 17 18 19 a 20 For Peer Review 21 Potential Energy 22 > 23 ZPE R 24 > (γγγH/γγγD)max 25 26 27 28 29 30 H-Transfer Coordinate 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 4 A pictorial representation of zero point energy and tunnelling 5 6 contributions to primary hydrogen isotope effects 7 8 9 10 R. A. More O’Ferrall 11 12 13 School of Chemistry and Chemical Biology, University College Dublin 14 15 16 17 Email: [email protected] 18 For Peer Review 19 20 21 ABSTRACT 22 The interpretation of primary hydrogen isotope effects in terms of isotopic sensitivities of 23 24 zero point in the reactants and transition state is reviewed. The reader is 25 26 reminded that the transition state for a hydrogen transfer reaction corresponds to the 27 28 position of maximum energy on the minimum energy path defined by the motion of the 29 heavy between which hydrogen is transferred. Tunnelling occurs through the 30 31 potential energy barrier separating the reactant and product channels of the potential energy 32 33 surface for this reaction. It is a consequence of rapid vibrational motion of the hydrogen 34 35 within a collision complex and the overlap of vibrational wave functions between reactant 36 37 and product sides of the barrier. Comparing zero point energies of a tunnelling complex 38 and a hydrogen bond confirms results of calculations indicating that the zero point energy 39 40 of the complex is likely to be less than that of the reactants. It follows that there should be 41 42 a compensation between contributions from zero point energy changes and tunnelling 43 44 which may explain why tunnelling rarely to very large isotope effects at ambient 45 . The isotopic sensitivity of the zero point energy of the transition state is 46 47 discussed in terms of variations in force constants of reacting bonds within a three-centre 48 49 model. A shift in isotopic sensitivity from the reaction coordinate mode to real stretching 50 51 vibration as the structure of the transition state becomes more reactant- or product-like, 52 53 which provides a basis for the Westheimer effect, contrasts with the behaviour of hydrogen 54 bonds, for which there is no change in isotopic sensitivity of the two stretching modes. 55 56 57 58 59 60 1

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1 2 3 INTRODUCTION 4 5 6 7 It is more than fifty years since Bigeleisen and Wolfsberg interpreted primary hydrogen 8 9 isotope effects in terms of differences in isotopically sensitive zero point energies between 10 1 11 reactants and transition state. In 1961Westheimer took that interpretation a step further by 12 identifying the stretching vibration associated with the transferring (not representing 13 14 the reaction coordinate) as a key factor controlling the magnitude of the isotope effect. 2 It 15 16 was argued that for a symmetrical transition state, in which the force constants to the 17 18 reacting hydrogen Foratom are the Peer same, there is Reviewno motion of hydrogen in this vibration and 19 consequently no change in zero point energy upon substitution of for hydrogen. 2-4 20 21 For a symmetrical transition state therefore the full effect of a zero point energy change in 22 23 the hydrogen stretching vibration of the reactant was expressed in the isotope effect, which 24 25 achieved a maximum value in the range kH/kD = 7-10 depending on the vibration 26 of the reactant and the extent to which isotopically sensitive bending vibrations were 27 28 cancelled between reactants and transition state. 4-6 For reactant and product like 29 30 transitions states kH/kD was presumed to approach 1.0 and a (small) equilibrium isotope 31 32 effect respectively. 33 34 35 The notion that primary hydrogen isotope effects possessed maximum values (the 36 37 Westheimer or Melander-Westheimer effect) 2-4 strongly influenced experimental 38 39 investigations in the following twenty years. A considerable effort was expended in 40 41 demonstrating that measurements of kH/kD for a series of related proton transfers, such as the 42 base-promoted ionisation ketones and nitroalkanes, did indeed exhibit a maximum when 43 44 plotted against equilibrium constants if these were varied over a sufficiently wide range. 7-9 45 46 47 48 Today greater interest is focused on the contribution of mechanical tunnelling 49 10-14 50 than zero point energy to hydrogen isotope effects, as witnessed by the symposium in 51 print of which this paper forms a part. Historically, tunnelling has been a focus of many 52 53 experimental and theoretical studies of hydrogen transfer reactions in the phase. 15-19 A 54 55 main concern of these papers was with quantum corrections to a based 56 57 calculation of reaction rate constants for semi-empirical potential energy surfaces arising both 58 59 60 2

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1 2 3 from tunnelling at low temperatures and reflection back of collision complexes crossing the 4 5 energy barrier at high temperatures. 20-24 Surprisingly there is little or no mention of the 6 7 Westheimer effect among the many studies of isotope effects for gas phase hydrogen transfer 8 9 reactions. This must be partly a consequence of the lack of series of reactions within which 10 11 the reactivity or equilibrium constants can be varied systematically in the manner possible for 12 organic reactions in solution. 13 14 15 16 While the main principles governing tunnelling were worked out in the years before 17 15-19 18 or soon after Westheimer’sFor treatment Peer of zero pointReview energies, the conclusions reached 19 were published in papers which sometimes included a significant mathematical treatment 20 21 of the topic. One widely read text book in which they were discussed was H.S. Johnston’s 22 15 23 “Gas Phase Reaction Rate Theory”. However, a principal focus of this text was on the 24 25 then new studies of reactive molecular scattering made possible by the development of 26 molecular beams and studies of , as well as the first serious 27 28 computer-based statistical simulations of reaction rates. Despite its importance, it is 29 30 perhaps not surprising that this did not reach a wider audience of experimental organic and 31 32 inorganic chemists. The purpose of the present paper is not to present new experimental or 33 34 theoretical results but to provide a straightforward introduction to tunnelling 35 complementary to the Westheimer-Melander treatment of zero point energy. 36 37 38 39 Early Treatments of Tunnelling 40 41 42 The name of R. P. Bell is associated with early treatments of tunnelling and his monograph 43 44 published in 1980 still provides the best introduction to the topic. 25 At a time before 45 46 computers gave fairly easy access to multidimensional potential energy surfaces, Bell 47 48 developed approximate treatments of tunnelling based on one dimensional energy barriers 49 50 and derived a quantum correction to the reaction coordinate mode of a hydrogen-transfer 51 transition state bearing a close analogy with the zero point energy correction to the 52 53 orthogonal stretching vibration 5,6 which was the main focus of Westheimer and Melander’s 54 55 treatment. These two vibrations are shown schematically for a saddle point and transition 56 57 state in Fig 1. The curvature of the potential energy surface along the reaction coordinate 58 59 60 3

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1 2 3 can be represented by a negative force constant f and associated with an imaginary 4 5 ≠ ≠ vibration frequency νL . The relationship between f and νL is shown in eqs 1 and 2 where 6 7 f refers to the curvature of a quadratic potential energy surface, E is the potential energy, r 8 ≠ 9 is a displacement along the reaction coordinate and the reduced mass. The value of νL is 10 11 a significant parameter characterising permeability to tunnelling. Thus a high curvature 12 ≠ -1 13 (large f and large imaginary value of νL [in units i sec ]) implies a ‘thin’ barrier and 14 consequently greater correction for tunnelling through the barrier relative to passage over 15 16 it. 17 18 For Peer Review 19 20 21 E 22 A----H----B 23 24 25 26 A----H----B 27 28 29 r 30 31 32 Figure 1 Reaction coordinate and ‘real’ vibrations at a transition state 33 34 35 2 2 36 d E/dr = -f (1) 37 ≠ 38 νL = (1/2 π)√f/ (2) 39 40 41 While the attempt to simplify and develop a mathematically tractable treatment of 42 43 tunnelling in the 1950s was understandable and necessary, in the long run an emphasis on 44 45 tunnelling as a correction to a transition state-based reaction rate may have been 46 47 misleading. As outlined below for the most commonly studied transfer of a hydrogen 48 between heavy atoms the principal pathways for tunnelling do not lie along the reaction 49 50 coordinate and the barrier subject to tunnelling does not correspond to the ‘classical’ 51 52 minimum energy path. Thus although an assessment of tunnelling based on this barrier 53 54 provides a convenient and pragmatic correction to a transition state-based rate constant for 55 relatively small amounts of tunnelling, it is perhaps not the best starting point for 56 57 understanding the origin of tunnelling. 58 59 60 4

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1 2 3 4 5 6 A PICTORIAL MODEL FOR TUNNELLING 7 8 9 To represent tunnelling qualitatively and pictorially it is helpful to recognise (a) that the one- 10 11 dimensional energy barrier normally shown for a reaction, which represents a cross section 12 along the minimum energy path (and reaction coordinate) of a potential energy surface, differs 13 14 from the barrier through which tunnelling occurs, and (b) that the atomic and molecular 15 16 leading to passage over this barrier also differ from those leading tunnelling. These 17 18 points can be demonstratedFor by Peer considering representative Review trajectories of reacting and 19 the potential energy barriers to their reaction. 20 21 22 23 Reaction coordinate for hydrogen transfer 24 25 26 The reaction coordinate for a hydrogen transfer corresponds to the collision of two ‘heavy’ 27 28 atoms (or molecules) to one of which a hydrogen is attached. In figure 2 the heavy 29 30 atoms are shown as A and B. In the course of a collision they approach each other quite 31 32 ‘slowly’ in comparison with the motion of the much lighter . The 33 34 difference in masses indeed allows us to a good approximation to separate the two motions. 35 Fig 2 is intended to show that on the time scale of the collision the hydrogen is vibrating 36 37 rapidly between the two atoms. 38 39 40 41 A BA B 42 43 44 45 Figure 2 Collinear collision complexes for hydrogen transfer: (a) early-stage (b) at 46 point of rebound 47 48 The transfer of a hydrogen atom (A-H + B -> A + H-B) is illustrated in Fig 3 on a 49 50 reaction coordinate diagram from which all but two potential energy contours bounding the 51 52 minimum energy path for reaction have been omitted. The position of the maximum in energy 53 54 along this pathway is indicated by the symbol ≠ representing the transition state. The rapid 55 56 motion of the hydrogen within the collision complex is indicated by the zig-zag path of a 57 58 59 60 5

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1 2 3 collision trajectory in which the vibration of A-H is dominant as the reactants approach each 4 5 other and of B-H as the products separate. 6 7 A-H + B 8 9 r A---H---B 10 BH 11 12 13 14 c' 15 16 17 A + H-B 18 For Peer Review 19 c 20 21 22 rAH 23 24 Figure 3 Reaction coordinate diagram showing the vibrational motion of hydrogen within 25 a collision complex leading to tunnelling 26 27 28 The collision trajectory shown does not pass through the transition state but cuts the 29 30 ‘corner’ at which it lies. The corner-cutting pathway is denoted cc' and corresponds to 31 32 motion of the hydrogen between the two heavy atoms while their positions remain fixed (i.e. 33 34 the increase in rAH equals the decrease in rBH ). As shown below, reaction complexes which 35 follow this pathway are tunnelling through an energy barrier. They to an increase in 36 37 reaction rate because the tunnelling occurs at a lower energy than that of the transition state. 38 39 The reaction path through the transition state surmounts a ‘semi-classical’ reaction barrier 40 41 (also shown below) without resort to tunnelling because at the transition state there is no 42 43 further barrier hindering transfer of the proton. The barrier is ‘semi-classical’ (or 44 ‘quasiclassical’ 26 ) because although it does not involve tunnelling it does include the zero 45 46 point energies of the transition state and reactants. 47 48 49 50 For a symmetrical (linear) transition state there is no appreciable motion of 51 hydrogen independently of the heavy atoms other than that constrained by stretching or 52 53 bending vibrations. For a collinear reactive collision following the minimum energy path 54 55 for reaction the transition state is at or close to the point of rebound of the reacting 56 57 molecules. From this point the hydrogen ‘rides on the back’ of the reagent B along the 58 59 60 6

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1 2 3 product energy channel having been delivered to the transition state attached to A. For a 4 5 proton transfer, because of the high energy of a bare proton, the combined bonding of A 6 7 and B to the proton along the reaction path is strongly conserved. 8 9 10 11 Potential energy surface s. 12 13 14 Most of the features of a qualitative description of tunnelling are contained in Fig 3. The 15 16 figure shows that tunnelling do not pass through the transition state and that the 17 27 18 barrier with which Fortunnelling isPeer associated differs Review from that along the reaction coordinate. 19 The difference in barriers may be shown by including the potential energy as contours as in 20 21 Fig 4. In this figure the curvature of the reaction coordinate is distorted to allow the use of 22 23 mass-weighted coordinates. The progress of a (classical) molecular collision can then be 24 25 represented by a single line tracing the trajectory of a ‘billiard ball’ directed on to the surface 26 with the appropriate energies of translational and vibrational motion. However, note that, as 27 28 usual, only linear configurations of the A-H-B collision complex are represented and that the 29 30 dashed line traces the minimum energy path for reaction up to and following displacement of 31 32 the hydrogen by tunnelling. 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Figure 4 Potential energy surface showing minimum energy and tunnelling pathways 52 53 54 55 The trajectory of a molecular collision leading to tunnelling is shown in figure 5 56 reproduced from a paper of Babamov and Marcus in 1981. 28 It can now be seen that the zig- 57 58 59 60 7

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1 2 3 zag pathway of the trajectory implies that at the limit of each vibration the hydrogen impacts 4 5 on the barrier separating the reactant and product channels. It is at the points of impact that 6 7 tunnelling is liable to occur. As implied above, tunnelling is easier when the barrier is thinner 8 9 and it can be seen that this is true when the collision complex is closer to the transition state. 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 26 27 28 29 30 Figure 5 Calculated trajectory (dashed line) of collision complex subject to tunnelling. 28 31 32 33 34 Whether or not in practice transfer of the hydrogen occurs by tunnelling will depend 35 on the and rate of reaction. At sufficiently high temperatures and for fast 36 37 reactions (with low energy barriers) a large proportion of collisions will possess sufficient 38 39 energy to surmount the ‘normal’ or heavy-atom reaction barrier without tunnelling. Other 40 41 things being equal tunnelling should be most important for slow reactions at low 42 temperatures or with high barriers. Good recent examples of both situations are provided 43 44 by double proton transfers in hydrogen-bonded carboxylic acid dimers discussed in this 45 46 issue by Horsewill 29 and by the intramolecular transfer of a proton of the carbene 47 o 48 hydroxymethylene to form (HOCH 2: -> O=CH 2) at 11 K in an argon 49 30 50 matrix. 51 52 53 The reaction trajectory in Fig 5 is an idealised one. The reaction rate is determined 54 55 by averaging over a great variety of collisions with different initial translational and, 56 57 indeed, rotational energies, as well as for non-collinear collisions. Moreover, the 58 59 60 8

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1 2 3 representation of the tunnelling trajectory by a straight line through the energy barrier 4 5 strictly implies that the proton moves fully independently of the heavy atoms. If this were 6 7 true the angle between the reactant and product channels of the surface would be zero. In 8 o o 9 practice typical angles are 15 for transfer of a hydrogen and 30 for the corresponding 10 26 11 transfer of deuterium between carbon, or nitrogen atoms. Moreover the sharp 12 curvature of the reaction path and relatively independent motion of the tunnelling hydrogen 13 14 in Figs 4 and 5 is characteristic of hydrogen transfer between heavy atoms. The behaviour 15 16 of the well studied isotope exchange of hydrogen molecules and hydrogen atoms is less 17 18 straightforward in Forthat the tunnelling Peer involves Review motion of three atoms implying a curved 19 trajectory in e.g. Fig. 3. 26,31 It is also an approximation to suggest that semi-classical and 20 21 tunnelling rates of reaction are fully separable. 26 22 23 24 25 Quantum mechanical description of tunnelling . 26 27 28 To complete a description of tunnelling an explanation of the barrier penetration is 29 30 required. This arises from the existence of zero point (and higher) energy levels for the 31 32 vibrating hydrogen atom in the reactant and product reaction channels. The vibrations 33 34 are associated with wave functions which allow extension of the bond to hydrogen 35 beyond its classical limits. In Fig 6 these zero point energies and the associated 36 37 vibrational wave functions are shown for a cross section through the surface along the 38 39 tunnelling reaction path cc' in Fig. 3 and extending across the reactant and product 40 41 channels to the edges of the potential energy surface. The cross section is a double 42 well potential which includes configurations of the collision complex in the reactant 43 44 (A-H---B) and product (A---H-B) reactant channels, for which the A-B distance is the 45 46 same but the position of the proton differs. The figure shows a symmetrical reaction in 47 48 which A and B are equivalent or nearly so. 49 50 51 52 53 54 55 56 57 58 59 60 9

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 6 Cross section along tunnelling coordinate cc' in Fig 3 showing overlap of zero 16 17 point vibrational wave functions 18 For Peer Review 19 20 Tunnelling between the two configurations is possible if there is significant overlap 21 22 of the wave functions. This can occur (a) if the barrier between the channels is sufficiently 23 24 thin (b) if the energies of the vibrations are closely matched. Strictly speaking the 25 symmetrical situation shown corresponds to ‘coherent’ tunnelling which is most commonly 26 27 associated with reversible phenomena such as inversion of a of ammonia. 28 29 Transfer of hydrogen atoms would normally be represented by incoherent tunnelling where 30 31 the energy levels differ (and the product is of lower energy than the reactant). Then 32 33 matching of energy levels may be achieved for reactions in solution by coupling with 34 solvent interactions, especially for proton transfer for which there will normally be a strong 35 36 solvent interaction with an atom bearing a positive or negative charge in the reactants or 37 38 products. However, even in the gas phase matching may be achieved at an appropriate 39 40 distance between the heavy atoms within a collision complex. 41 42 43 The energy levels in Fig 4 correspond most obviously to zero point energies, but the 44 45 discussion appears not to be qualitatively changed if some or many reaction trajectories 46 47 involve vibrational excitation. In principle such excitation should favour tunnelling, as it 48 49 allows access to a thinner barrier and more favourable overlap of wave functions. 50 Moreover for an unsymmetrical barrier and highly exothermic reaction it is likely that 51 52 hydrogen transfer will occur from the zero point of the reactant to a 53 54 vibrationally of the product, even for reaction over the (semi) classical energy 55 35 56 barrier. 57 58 59 60 10

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1 2 3 It should be added that most computational treatments of hydrogen transfer include 4 5 tunnelling not as the phenomenon illustrated in Fig. 6 but as a hydrogen atom 6 7 impinging on an energy barrier, as considered by Bell. Much of the above discussion, 8 9 which emphasises the rather than wave character of the proton, is more appropriate 10 11 to that concept. Strictly speaking Fig 6 represents a condition of resonance in which the 12 hydrogen is delocalised between reactant and product states and relaxation of the heavy 13 14 atoms of the collision complex relocalises it in its reactant or product structure. 15 16 17 18 Good introductionsFor to thePeer different ways Review of analysing tunnelling are provided in the 19 introductions to several papers in this symposium including references 26, 29 and 32. 20 21 Also pertinent are papers by Siebrand 33 and Kutznetsov. 34 A fundamental problem is to 22 23 sum contributions over a range of energies and structures of collision complexes. While 24 25 rate constants based on transition state theory provide a good approximation to quantised 26 trajectory calculations on potential energy surfaces when tunnelling is not taken into 27 28 account it has been much more difficult to find a correspondingly convenient 29 30 approximationfor calculating a rate constant when tunnelling is important. 31 32 33 34 Tunnelling by deuterium can be expected to be less effective than for hydrogen 35 because the larger mass of deuterium implies a more localised vibrational 36 37 with less overlap between configurations of collision complexes in reactant and product 38 39 valleys. In addition, the lower zero point energy of a bond to deuterium implies that the 40 41 vibrating atom is impinging on the barrier to atom transfer at a lower energy and hence 42 greater barrier width than the hydrogen. 43 44 45 46 Finally, two more representations of potential energy surfaces for hydrogen transfer 47 48 between heavy atoms are shown in Figs 7 and 8. Fig 7 shows a projection of the double well 49 50 potential of Fig 6 on a reaction coordinate diagram based on Fig 4 including an idealised 51 reaction trajectory corresponding to tunnelling between the wells. Also shown is the 52 53 disappearance of the double well, and hence of a barrier to proton transfer, at the transition 54 55 state (the cross section for the single well should be at right angles to that for the double well 56 57 58 59 60 11

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1 2 3 for the symmetrical reaction shown). The direction of a proton transfer trajectory with the 4 5 positions of the A and B atoms fixed is shown by the arrow labelled ‘H-vibration’. 6 7 8 9 10 11 12 13 14 15 16 17 18 For Peer Review 19 20 21 22 23 24 25 Figure 7 Projection of cuts across a potential energy surface onto a skewed reaction 26 27 coordinate for hydrogen transfer at the transition state and along a tunnelling coordinate 28 29 30 Fig 8 is a schematic representation of the classical barrier for a heavy atom collision 31 32 leading to transfer of a hydrogen without tunnelling. Again note that there is no 33 34 independent movement of the hydrogen along this reaction coordinate. The hydrogen is 35 36 compressed in the collision between the heavy atoms and in principle rebounds with either 37 A or B with approximately equal probability for a collision trajectory reaching the 38 39 transition state. 40 41 42 43 A---H---B 44 45 46 47 48 A + H-B 49 50 51 H transfer 52 rB-H coordinate 53 A-H + B 54 55 56 57 Figure 8 Classical barrier to reaction and a ‘non-tunnelling’ reaction trajectory 58 59 60 12

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1 2 3 TUNNELLING AND ZERO POINT ENERGY 4 5 6 7 The barrier to the reaction in Fig. 8 is classical rather than semi-classical because no zero 8 9 point energies are shown. In practice collision complexes do have zero point energy. For 10 11 the reaction A-H + B -> A + H-B, the zero point energy of a reactant-like complex should 12 approximate that of the A-H vibration and that of a product-like complex the B-H vibration. 13 14 The zero point energy can be seen as providing a variable ‘flat bottom’ above the valley 15 16 floors of a potential energy surface corresponding to the minimum energy pathway between 17 18 reactant and products.For It is then Peer interesting toReview ask how the magnitude of this zero point 19 energy varies with the geometry of collision complexes based on an analogy with hydrogen 20 21 bonds or from calculations for theoretical or semi-empirical potential energy surfaces. 18 22 23 24 25 Studies of O-H----O hydrogen bonds in allow access to a wide range of 26 geometries. 35,36 In general, hydrogen bonding reduces the frequency of the covalent O-H 27 28 stretching vibration and introduces a low frequency vibration which corresponds to a 29 30 hindered translation in the limit of a very weak hydrogen bond. For the few symmetrical 31 - 37,38 32 hydrogen bonds known, such as HF 2 , the vibrations conform to the characteristic 33 34 symmetric and asymmetric stretching vibrations we associate with other linear three centre 35 symmetrical molecules such as CO . 36 2 37 38 39 The changes in geometry for different strengths of hydrogen bonds have been 40 39,40 41 considered to map the reaction coordinate of a proton transfer reaction. In a similar way 42 we might take the changes in vibration frequency as mapping the variation in isotopically 43 44 sensitive zero point energy along this coordinate. If the vibration of alcohols 45 46 in the gas phase and of carboxylic acid-carboxylate anion complexes in the solid state offer 47 -1 48 suitable guides we can say that this frequency decreases from 3600 cm for a covalent O-H 49 -1 50 bond to 700cm for hydrogen symmetrically bound between two oxygen atoms. These 51 measurements apply to equilibrium configurations of hydrogen bonds, but there is some 52 53 indication that ‘dynamic’ compression of bond distances in molecular collisions affect the 54 41 55 vibration frequencies similarly. While it is a large step from equilibrium measurements 56 57 for hydrogen bonds between electronegative atoms to a collision complex for a proton or 58 59 60 13

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1 2 3 hydrogen transfer from a carbon atom, it seems reasonable to suppose that a similar loss of 4 5 zero point energy accompanies compression of (say) a C-H bond in a molecular collision. 6 7 8 9 If we consider the relationship between contributions from zero point energy and 10 11 tunnelling to an isotope effect it has sometimes been supposed that the two effects augment 12 each other. However, if we examine limiting cases for which, on the one hand, all 13 14 collision complexes pass over an energy barrier and, on the other, all hydrogen atoms 15 16 tunnel through it, it seems more reasonable to conclude that the two contributions are 17 18 complementary. For Peer Review 19 20 21 Thus, in the first limit the isotope effect is given by eq 3, in which ZPE ≠ and 22

23 ZPE R are differences in H and D isotopic zero point energies in the transition state and 24 25 reactants respectively, kT is the , and for simplicity contributions from 26 27 changes in isotopic partition functions are neglected . 28 29 30 -(ZPE ≠ - ZPE )/ kT k /k = e R (3) 31 H D 32 33 34 For the second limit we suppose that reaction occurs from the bottom of the reactant valley 35 36 of a potential energy surface and that the zero point energy of the reacting collision 37 38 complex is not appreciably perturbed from that of the reactants. The isotope effect is then 39 expressed as in eq 4 by the ratio of tunnelling (transmission) coefficients for the H and D 40 41 , γH/γD. The ratio can be treated as an upper limit [(γH/ γD)max ] because the width 42 43 of the proton transfer barrier for a reactant-like collision complex is large and because the 44 45 (zero point) energy separation between H and D complexes is at a maximum. The 46 47 difference in zero point energy does not contribute directly to tunnelling but it implies a 48 narrower effective barrier for hydrogen than deuterium. This is illustrated in Figure 9a. 49 50 51 52 kH/kD = ( γH/ γD)max (4) 53 54 55 56 57 58 59 60 14

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1 2 3 4 5 6 7 8 b 9 10 ZPE 11 C 12 13 14 15 16 a

17 Potential Energy 18 For Peer Review> 19 ZPE R > (γγγH/γγγD)max 20 21 22 23 24 H-Transfer Coordinate 25 26 27 28 Fig 9 Energy barriers for hydrogen tunnelling: (a) for a reactant-like collision 29 complex (A-H-----B) and (b) a collision complex closer to the transition state (A--H---B) . 30 31 The hydrogen transfer coordinates for the complexes correspond to cross sections at 32 33 different points on the ‘heavy atom’ coordinate of Fig 8. As indicated in fig 8 (and figs 4, 34 35 5 and 7) the energy maximum for barrier (b) should be lower than that for (a). For clarity 36 in a one-dimensional representation energy curve (b) is displaced vertically. 37 38 39 40 More generally the isotope effect will reflect a combination of zero point energy 41 42 and tunnelling contributions corresponding to a collision complex which is intermediate in 43 44 structure between reactants and transition state. H and D zero point energies and the 45 barrier to hydrogen transfer for such a complex are shown in Fig 9b. Now the isotope 46 47 effect depends not only on tunnelling but on the difference in zero point energies of the 48 49 complex and reactant, ZPE C - ZPE R. The contribution of the two factors is expressed in 50 51 eq 5. Because the barrier to hydrogen transfer for the complex is narrower and the 52 difference in energy of H and D complexes is smaller, the tunnelling contribution to the 53 54 isotope effect, γ H/γD, is less than for a more reactant-like complex. However, this reduced 55 56 tunnelling contribution is offset by a favourable zero point energy effect. 57 58 59 60 15

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1 2 3 4 5 -(ZPE - ZPE ) kH/kD = ( γH/ γD) e C R /kT 6 7 8 9 In practice rate constants represent many reaction trajectories with tunnelling 10 11 occurring from a range of structures of collision complexes. However, if tunnelling and 12 zero point energy contributions vary between the above limits it is clear that the zero point 13 14 energy term should be smaller where γH/ γD is larger and vice versa. In other words one can 15 16 envisage a pay off between zero point energy and tunnelling contributions to isotope 17 18 effects, so that as tunnellingFor increasesPeer the zero Review point energy effect is reduced. The 19 20 compensation between the two effects could offer an explanation as to why large primary 21 hydrogen isotope effects (i.e. > 10) at ordinary temperatures (near 25 o) are not common 22 23 and that measured isotope effects remain within the semiclassical limits implied by zero 24 25 point energy changes even where the contribution of tunnelling is appreciable. 26 27 28 29 This argument based on hydrogen bonds is an intuitive one, but it is consistent with 30 18 calculations for potential energy surfaces, e.g. of the H + H 2 isotope exchange reaction. 31 32 Indeed changes in zero point energies along the reaction coordinate (and corresponding 33 34 differences in vibrational energies between isotopic collision complexes) are implicitly taken 35 36 into account in tunnel corrections based on such surfaces. This variation in zero point 37 energies is similar to, but not quite the same as, that used by Westheimer to explain the 38 39 dependence of isotope effects on the location of the transition state for hydrogen transfer 40 41 along a reaction coordinate. Moreover, as indicated below, and by measurements of 42 43 equilibrium isotope effects, the net change in isotopic zero point energy accompanying 44 formation of a hydrogen bond is likely to be less than that for a transition state. If that is the 45 46 case, the extent to which tunnelling is offset by zero point energy changes may be quite 47 48 limited. 49 50 51 52 53 54 55 56 57 58 59 60 16

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1 2 3 Zero point Energy at the Transition State 4 5 6 7 At the transition state, zero point energy contributes to the reaction barrier and semi- 8 9 classical isotope effects arise from a difference of H and D zero point energies between the 10 11 reactants and transition state. While it seems possible to estimate zero point energies 12 associated with the reaction channels leading to the transition state by analogy with 13 14 hydrogen bonding, the situation for the transition state itself is different 15 16 17 18 At the transitionFor state thePeer potential energy Review surface for a reaction corresponds to a 19 saddle point. The quadratic region of the saddle point may be analysed in terms of the 20 21 stretching vibrations of a linear three centre model with force constants to the reacting 22 23 hydrogen f1 and f2 and interaction force constant f12 as depicted in 1. When the model 24 25 represents a transition state the curvature along the reaction coordinate must be negative 26 and the frequency corresponding to this curvature is then imaginary. This is true when f 27 12 28 > √f1f2. 29 30 31 32 f1 f2 33 A - - - - - H ------B 34 f12 35 36 1 37 38 39 f f 40 For a symmetrical transition state with 1 = 2 there is no zero point energy 41 difference between the notional H and D frequencies corresponding to motion orthogonal 42 43 to the reaction coordinate and the contribution of this zero point energy to the reaction 44 45 barrier is the same for both isotopes. This is similar to the situation for a hydrogen bond 46 47 except that the isotopically sensitive frequency is now an imaginary value corresponding to 48 the reaction coordinate. 49 50 51 52 To consider the isotopic sensitivity of the ‘real’ vibration and reaction coordinate 53 54 mode of the transition state it is relevant to be reminded that their combined sensitivity is 55 controlled by the product rule. 1,42 In a single for a triatomic collinear collision 56 57 complex this is expressed in eq 6, in which ω and ω ≠ represent real and imaginary 58 59 60 17

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1 2 3 frequencies (in cm -1) of the two stretching vibrations and m and M are masses of hydrogen 4 5 isotopes and isotopic collision complexes respectively. 6 7 8 ≠ ½ ½ 9 ωHωH MH mH ______= ______10 (6) 11 ≠ ½ ½ ωDωD MD mD 12 13 14 15 For hydrogen bonds we expect isotopic sensitivity to be confined to a single vibration, 16 17 characteristically the vibration of higher frequency. However for a sufficiently large 18 interaction constant,For i.e. as f Peerapproaches ( f +Review f )/2, both vibrations become isotopically 19 12 1 2 20 sensitive. Indeed, as f12 approaches ( f1 + f2)/2 closely (and f1 and f2 have similar values) 21 22 isotopic sensitivity is again confined to a single vibration, but this is now the vibration with 23 24 lower frequency. As f12 becomes equal to √f1f2 this frequency passes through zero and 25 f √f f 26 increases again as an imaginary frequency (for 12 > 1 2). 27 28 29 For a symmetrical transition state f1 = f2, and ( f1 + f2)/2 = √f1f2. In this case, as we 30 31 have seen, there is no motion of hydrogen in the real stretching vibration of the transition 32 33 state. For unsymmetrical transition states, the force constants f1 and f2 differ. Then ( f1 + 34 f )/2 is greater than √f f , and this inequality becomes more pronounced as the disparity 35 2 1 2 36 between f 1 and f 2 increases. A consequence is that there is a shift in isotopic sensitivity 37 38 from the reaction coordinate mode to the real stretching vibration of the transition state. 39 40 Thus the small isotope effects envisaged by Westheimer for reactant- or product-like 41 transition states arise both from an increasing magnitude and increasing isotopic sensitivity 42 43 of the real vibration and associated zero point energy at the transition state. 44 45 46 47 An extended analysis of the influence of stretching and interaction force constants 48 49 on both isotopic sensitivity and bond displacements for the vibrations of a three centre 50 model has been provided by Albery. 43 He points out that variations in isotopic sensitivity 51 52 of the zero point energy are associated with a change in reaction coordinate from the 53 54 isotopically insensitive translational approach of the reactants for a highly reactant-like 55 56 transition state to a practically independent (and isotopically sensitive) motion of the 57 hydrogen atom in a symmetrical transition state. As noted above, this behaviour contrasts 58 59 60 18

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1 2 3 4 with that of hydrogen bonds for which the larger stretching frequency νs is expected to 5 show full isotopic sensitivity for all degrees of symmetry or asymmetry. 6 7 8 9 The relationship between force constants, masses and frequencies for these 10 15 11 vibrations is conveniently summarised in an appendix to H. S. Johnston’s book and 12 reproduced below. 13 14 15 16 Gas Phase and Solution 17 18 For Peer Review 19 20 In principle the discussion provided above applies to gas phase reactions, which 21 normally involve transfer of hydrogen atoms. Most studies of isotope effects and tunnelling 22 23 apply to proton or transfer reactions in polar solvents, especially . 1-13 24 25 Generally, it has been supposed that collision complexes of reactants in solution or for 26 27 enzymes do not differ fundamentally from those in the gas phase, although the translational 28 motions associated with a molecular collision in the gas phase may be replaced by a weak 29 30 stretching vibration (hindered translation) of a hydrogen bond in solution. However, the 31 32 potential energy surface for the reaction will be strongly influenced by the solvent or protein 33 34 environment, especially for ionic reactions. 35 36 37 A reasonable view of interactions of reactants with the solvent is that reorientation of 38 39 solvent molecules to accommodate developing charges accompanying a reaction occur prior 40 41 to or following the -making and bond-breaking step of a reactive molecular 42 43 collision. However, within the time frame of a ‘heavy atom’ collision (or hindered translation) 44 relaxation of solvent molecules can occur along preformed hydrogen bonds or orientated lone 45 46 pair interactions with (e.g.) a centre of developing positive charge. 47 48 49 50 A point discussed for tunnelling in solution and enzymatic reactions is the influence of 51 fluctuating solvent interactions in achieving a matching of vibrational energy levels between 52 53 reactant and product reaction channels of the potential energy surface (fig 3) thereby 54 55 facilitating tunnelling . This falls outside the scope of the present discussion but is addressed 56 57 in other papers of this symposium in print. 58 59 60 19

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1 2 3 Conclusion 4 5 6 The main purpose of this analysis is to provide a qualitative description of a plausible 7 8 relationship between contributions to primary hydrogen isotope effects from zero point energy 9 10 and tunnelling. The Westheimer-Melander effect has provided a historic guide for 11 12 experimental investigations of hydrogen isotope effects. Marrying this zero point energy- 13 based analysis to a not too naïve representation of tunnelling might be considered one 14 15 challenge presented for this symposium. 16 17 18 For Peer Review 19 It can certainly be claimed that the revival of interest in tunnelling has been stimulated 20 45 21 by experimental measurements for enzyme-catalysed reactions. The measurements have 22 raised the question of whether enzymes might facilitate tunnelling. With some notable 23 24 exceptions 45-47 it seems fair to say that this question has been addressed more actively by 25 26,33,48 26 computational or theoretical chemists than by organic or inorganic chemists undertaking 27 28 measurements of isotope effects for non-enzymatic reactions. It would be a useful upshot of 29 the symposium if more experimentalists were encouraged to take an interest in the field. 30 31 32 33 NOTE ON THREE CENTRE MODEL ( 1) AND HYDROGEN BONDS 34 35 36 37 Convenient relationships between frequencies of stretching vibrations, isotopic masses and 38 force constants for a linear three atom model for a hydrogen bond or hydrogen transfer 39 40 transition state (1 above ) have been provided by H. S. Johnston 15 and are reproduced in eqs 4 - 41 2 2 2 42 6 below. The two vibration frequencies are represented by λ1 and λ2, where λ = 4 π c ω and 43 44 c is the velocity of ( ω = ν/c) . As a minor simplification the masses of A and B are taken 45 46 as equal ( m) and presumed to be at least ten times greater than the mass of hydrogen mH. 47 48 49 2 ½ λ1 = ½[ B + ( B - 4 C) ] 50 2 ½ 51 λ2 = ½[ B - ( B - 4 C) ] 52 53 B = ( f1 + f2)/ m + ( f1 + f2 – 2 f12 )/ mH 54 2 2 55 C = ( f1f2 – f12 )(2 m + mH)/ m mH 56 57 58 59 60 20

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1 2 3 m C m 4 It is apparent that when is sufficiently large and λ2 approach zero. Usally >> 5 m and λ is then very much less than λ . The value of λ becomes comparable to λ only 6 H 2 1 1 2 7 when f12 approaches ( f1 + f2)/2 and B become small. Values of C and λ2 also fall to zero when 8 ≠ 9 f12 = √f1f2. When f12 > √f1f2, C and λ2 become negative and the frequency ωL corresponding 10 11 to λ2 is imaginary. 12 13 14 15 When the interaction constant f12 is not too large it is evident that B is isotopically 16 17 sensitive ( BH/BD -> mH/mD) because m >> mH. In the normal case that C is also small only λ1 18 will show appreciableFor isotopic Peer sensitivity. ThisReview is typical of hydrogen bonds. Thus for the 19 H 20 symmetrical hydrogen bond of the bifluoride the stretching frequencies ωa = 1452 and 21 H -1 - D D -1 - 22 ωs = 600 cm for HF 2 , and ωa = 1040 and ωs = 600 cm for DF 2 are well fitted by the 23 5 24 (harmonic|) force constants f1 = f2 = 2.32 and f12 = 1.71 x 10 dynes/cm. 25 26 27 In the case of HF - the interaction force constant f is quite large. However, it is only as f 28 2 12 12 29 approaches ( f1 + f2)/2 that relative magnitudes and typical isotopic sensitivities of λ1 and λ2 30 31 break down. Then B becomes isotopically insensitive and there is a switch in sensitivities and 32 33 relative magnitudes between λ1 and λ2. At extreme values of f12 ≥ √f1f2, C = 0 or is negative 34 35 and the three centre model now represents a transition state. 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 21

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1 2 3 4 5 References 6 7 8 9 1) J. Bigeleisen and M. Wolfsberg, Adv. Chem. Phys. 1958 , 1, 15-76 10 11 2) F. H. Westheimer, Chem. Revs. 1961 , 61 , 265 12 3) L. Melander, Reaction Rates of Isotopic Molecules , Ronald, New York, 1960 13 14 4) L. Melander and W. H. Saunders, Jr., Isotope Effects on Reaction Rates , John 15 16 Wiley, New York, 1980 17 18 5) R. A. MoreFor O’Ferrall inPeer Proton Transfer Review Reactions , E. F. Caldin and V. Gold, eds., 19 Chapman Hall, London, 1975 . 20 21 6) R. P. Bell, The Proton in Chemistry , 2 nd ed., Chapman Hall, London, 1973. 22 23 7) R. P. Bell and D. M. Goodall, Proc. Roy. Soc. A 1966 , 294 , 273. 24 25 8) D. J. Barnes and R. P. Bell, Proc. Roy. Soc. A 1970 , 318 , 421. 26 9) F. G. Bordwell and W. J. Boyle, Jr., J. Am. Chem. Soc. 1975 , 97 , 3447. 27 28 10) A. Kohen and H. H. Limbach, Eds. Isotope Effects in Chemistry and Biology , 29 30 Taylor and Francis, CRC Press, Boca Raton, Florida, 2006 . 31 32 11) F. E. Romesberg and R. L. Schowen, Adv. Phys. Org. Chem. 2004 , 39 , 27. 33 34 12) N. S. Scrutton and R. K. Allemann, Eds., in Enzyme-Catalysed 35 Reactions , Royal Society of Chemistry, Cambridge, 2008 . 36 37 13) Z. D. Nagel and J. P. Klinman, Chem. Rev. 2006 , 106 , 3055. 38 39 14) A. M. Kuznetsov and J. Ulstrup, Can. J. Chem. 1999 , 77 , 1085. 40 41 15) H. S. Johnston, Gas Phase Reaction Rate Theory , Ronald, New York, 1966 . 42 16) R. E. Weston, Jr., J. Chem. Phys . 1959 , 31 , 892. 43 44 17) H. S. Johnston and D. Rapp, J. Am. Chem. Soc. 1961 , 83 , 1. 45 46 18) D. G.Truhlar and A. Kupperman, J. Am. Chem. Soc. 1971 , 93 , 1840. 47 48 19) Marcus, R. A. and Coltrin, M. E., J. Chem. Phys. 1977 , 67 , 2609 49 50 20) Li-Ping Ju, Ke-Li Han and J.Z.H. Zhang, J. Comput. Chem. 2008 , 30 , 305. 51 21) B.C. Garrett, Theor. Chem. Acc. , 2000 , 103 , 200. 52 53 22) D. G. Truhlar, B. C. Garrett and S. J. Klippenstein, J. Phys. Chem. 1996 , 100 , 12771. 54 55 23) W. H. Miller, Acc. Chem. Res. 1993 , 26 , 174. 56 57 24) D. G.Truhlar, W. L. Hase and J. T. Hynes, J. Phys. Chem. 1983 , 87 , 2684. 58 59 60 22

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1 2 3 25) R. P. Bell, The Tunnel Effect in Chemistry , Chapman and Hall, London 1980 . 4 5 26) D. G. Truhlar, J. Phys Org. Chem. 2010 , 23 , paper in this issue. 6 7 27) M. M. Kreevoy, D. Ostovic and D. G. Truhlar, J. Phys. Chem. 1986 , 90 , 3766. 8 9 28) V. K. Babamov and R. A. Marcus, J. Chem. Phys. 1981 , 74 , 1790. 10 11 29) A. J. Horsewill, J. Phys Org. Chem. 2010 , 23 , paper in this issue. 12 30) P. R. Schreiner, H. P. Reisenauer, F. C. Pickard, A. C. Simonett, W. D. Allen, E. 13 14 Matyus and A. G. Csaszar, 2008 , 453 , 906. 15 16 31) R. A. Marcus, J. Phys. Chem. 1979 , 83 , 204. 17 18 32) H. H.Limbach,For K. B. SchowenPeer and R. ReviewL. Schowen, J. Phys Org. Chem. 2010 , 23 19 paper in this issue. 20 21 33) W. Siebrand, T. A. Wildman and M. Z. Zgierski, J. Am. Chem. Soc., 1984 , 106 , 22 23 4083, 4089. 24 25 34) E. D. German, A. M. Kuznetsov and R. Dogonadze, J. Chem Soc. Faraday Trans. 26 II 1980 , 76 , 1128. 27 28 35) T. Steiner, Angew. Chem. Int. Ed. 2002 , 41 , 48. 29 30 36) E. Libowitzky, Monatshefte Chemie 1999 , 130 , 1047. 31 32 37) A. Novak, Struct. Bonding 1974 , 18 , 177. 33 34 38) V. Spirko, A. Cajehan and G. J. Dickensen, Chem. Phys. 1991 , 45 , 151. 35 39) J. D. Dunitz, Phil. Trans. R. Soc. Lond. B 1975 , 272 , 99. 36 37 40) P. Schuster, G. Zundel and C. Sandorfy, Eds., The Hydrogen Bond , Vol 2, North 38 39 Holland, Amsterdam, Chapter 8, 1976 . 40 41 41) S. Bratos, J.-C. Leicknam and S. Pommeret, Chemical 2009 , 359 , 53 42 42) G. Herzberg, Infrared Spectra of Polyatomic Molecules , Van Nostrand, New York, 1945 43 44 43) W. J. Albery, Trans. Faraday Soc. 1967 , 63 , 200 45 46 44) M. J. Knapp, K. Rickert and J. P. Klinman, J. Am. Chem. Soc. 2002 , 124 , 3865. 47 48 45) N. Backstrom, N. A. Burton S. Turega and C. I. F. Watt, J. Phys. Org. Chem. 2007 , 49 50 21 , 603 51 46) M. Barroso, L. G. Arnaut, S. J. Formosinho, J. Phys. Org. Chem. 2010 , 23 , paper 52 53 in this issue. 54 55 47) K. M. Doll and R. G. Finke, Inorg. Chem. , 2003 , 42 , 4849. 56 57 48) M. H. M. Olsson, P. E. M. Siegbahn and A. Warshel, J. Am. Chem. Soc. 2004 , 126 , 2820. 58 59 60 23

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