Quantum Theory of Kinetic Isotope Effects in Proton Transfer Reactions B Y NINA BRUNICHE-OLSEN AND JENS ULSTRUP" Chemistry Departments A and B, Buildings 207 and 301, The Technical University of Denmark, 2800 Lyngby, Denmark Received 10th April, 1978 Within the framework of multiphonon electron and atom group transfer theory we have calculated the kinetic isotope effect of proton transfer reactions in homogenous solutions using both harmonic and anharmonic potentials for the proton. The calculations can reproduce all the important features of the isotope effect, such as the maximum for the zero free energy change, and the higher activation energy and Arrhenius pre-exponential factor for the heavier isotope. Quantitative agreement with several experimental data relating to the dependence of the isotope effect on the free energy of reaction and the temperature is furthermore obtained for plausible values of the two parameters required, i.e., the solvent reorganization energy and the proton transfer distance. 1.INTRODUCTION A fundamental result of the theory of electron and atom group transfer processes is that nuclear tunnelling is expected when sufficiently high vibration frequencies are associated with the nuclear modes undergoing reorganization.'. However, unambiguous evidence for nuclear tunnelling is usually difficult to obtain, i.e., the majority of experimental results concerning the kinetics of elementary processes in condensed phases can be explained both by the elaborate quantum theory of chemical processes and by much simpler semiclassical approaches.Nuclear tunnel effects in chemical processes are, however, manifested in low- temperature proce~ses,~-~ in strongly exothermic processes and in particular in homogeneous and heterogeneous proton transfer reactions. '-' Evidence for proton tunnelling is thus usually based on : (a) " unusually " large absolute values of the kinetic isotope effect, kH/k,, where kH and k,, are the rate constants for proton and deuteron transfer, respectively ; (b) different activation energy for proton and deuteron transfer (i.e., a larger difference than the difference in initial state vibrational zero- point energies) ; (c) pre-exponential Arrhenius factors which are smaller for the proton transfer and ( d ) a temperature dependent apparent activation energy for the rate constant or the isotope effect.In the semiclassical theory of proton transfer reactions, with tunnel corrections, major attention is given to motion of the proton along a stretching coordinate. This motion is assumed to provide both the activation energy and the tunnel corrections by proton tunnelling near the barrier tops* In contrast, the key results of the quantum theory of multiphonon proton transfer reactions are that the activation energy is provided by excitation of a multitude of low-frequency intramolecular and solvent modes, whereas the proton in general proceeds from its initial to its final state ground vibrational level by tunne1ling.l A considerable amount of evidence for the latter view has recently been provided by studies of the electrochemical hydrogen evolution reaction on mercury and other metal electrodes of low hydrogen adsorption energy.This evidence can be summarized as follows :lo 205206 THEORY OF PROTON TRANSFER REACTIONS (A) The transmission coefficient is also low when the reaction proceeds under barrierless conditions, i.e. when the transfer coefficient, a, is unity. (B) The second step in the overall process, i.e., the electrochemical desorption, also displays a substantial isotope effect when this step is activationless, i.e., when a -+ 0. Both of these effects are incompatible with the semiclassical view, but not with the quantum theory of proton transfer reactions. (C) The pre-exponential factor of the electrochemical rate constant decreases with increasing adsorption energy of the hydrogen atom at the metal electrode.This effect is understandable on the basis of quantum theory, whereas the semiclassical theory would predict the inverse relationship. (D) Comparison of the activation parameters of the hydrogen evolution reaction in water and acetonitrile shows that the solvent exerts a pronounced effect on the activation energy of the process, whereas the nature of the depolarizer ( i e . , H30+ or CH3CNH+) largely determines the value of the pre-exponential factor. Studies of kinetic isotope effects in homogeneous and heterogeneous proton transfer reactions have thus provided the most comprehensive evidence for nuclear tunnelling. In the following we shall present a quantitative application of the formalism of multiphonon atom group transfer theory to kinetic isotope effects in such p r o c e ~ s e s .~ ~ - ~ ~ In particular we shall extend previous results based on a harmonic oscillator representation of the proton 12* l3 by the application of Franck- Condon nuclear overlap factors also for Morse and squared hyperbolic tangent potentials, these being more realistic representations of the progon stretching and bending modes, respectively. 2. SUMMARY OF MULTIPHONON ATOM GROUP TRANSFER THEORY The rate constant of the elementary atom group transfer process is usually expressed in the nonadiabatic limit and by the time evolution of zero-order Born-Oppenheimer vibronic states corresponding to the atom group being localized on the donor and acceptor. 1s Moreover, the high-frequency modes associated with the transferring atom group are essentially deconvoluted from all other intramolecular and (low- frequency) medium modes.The latter may be strongly or weakly coupled to the reaction centre. If the coupling furthermore is linear, the resulting rate expression shows a gaussian or lorentzian free energy dependence, respectively,16 where we shall apply the former limit only, which is appropriate for most proton transfer reactions in condensed media. Finally, for reasons discussed previously,l* 11* l 5 we shall consider proton transfer between immobile donor and acceptor fragments, the relative distance and orientation of which are characterized by a vector, R, i.e., we shall assume that the mass of the proton is so much smaller than the masses of the fragments that the latter remain stationary during the proton transfer.We shall thus basically apply the following expression for the rate constant k 1 9 ' 9 l1 to k = Q(R)W(R) dR RdU where W(R) = [ V(R)12(n/kTA2Es)*Z-1 exp (-PQ x v w E, is here the reorganization energy of the medium and all other low-frequency modes, AE the free energy of reaction [the energy gap, equal to -kTln (KA/KB), where KA and KB are the acid dissociation constants of the proton donor and acceptor,N. BRUNICHE-OLSEN AND J . ULSTRUP 207 respectively], EY and E? the proton vibrational energy levels in the initial and final state, respectively, corresponding to the vibrational quantum numbers v and w, Z the statistical sum of the transferring proton modes in the initial state, and p = kT, where k is Boltzmann's constant and T the absolute temperature.Furthermore, Sv,,v = [(@'14u)lz is the Franck-Condon nuclear overlap factor of the total proton (stretching and bending) nuclear wave functions in the initial (4;) and final ($r) state, V(R) the electronic exchange integral, and the function @(R) expresses the probability that the particular configuration characterized by R is achieved. a@) is the quantum statistical distribution function of the reactants. However, for the sake of simplicity we shall introduce the plausible assumption that the motion of the reactants as a whole is classical, at least up to the value of R = Rmi,, and that O(R) can therefore be represented by the form @(R) = exp [-PU(R)I (3) where U(R) is the potential energy of interaction between the reactants.We shall furthermore assume that U(R) is the same in the initial and final states. We noticed previously l5 that most proton transfer reactions are likely to belong to the category of partially adiabatic processes.'* l7 The electronic interaction between donor and acceptor fragments is sufficiently strong that only the lowest (electronically) adiabatic potential energy surface needs to be invoked. On the other hand, the probability of proton transfer is low, i.e., the proton must penetrate a large barrier, even when the system has acquired sufficient energy to reach the saddle point of the reaction hypersurface with respect to the classical (medium) modes. All fundamental implications of the nonadiabatic scheme represented by eqn (1)-(3) then remain valid, but it is convenient to rewrite the equations in a form which no longer involves the electronic factor,l* l7 i.e.Weff W(R) = 1 -Su,w(R) exp (-BE!) exp [-/~(E,+AE+E,"-EY)~/~E,] (4) 0 w 2x where uCff is the effective frequency of the whole classical (medium) system.18t Eqn (4) essentially constitutes a semiclassical formulation of the rate expression for proton transfer reacti0ns.l it contains an activation factor predominantly determined by the medium and other low-frequency modes, and a gre-exponential factor consisting of the medium frequency and the Franck- Condoii nuclear overlap (tunnel) factor of the transferring proton or deuteron. Strictly speaking, the nonadiabatic activation energy of eqn (4) should be diminished by the resonance splitting energy at the reaction hypersurface.l* l7 However, since our basic conclusions below would not be qualitatively affected by this correction, and in view of our lack of quantitative information about its magnitude, we shall not include it in subsequent analysis.U(R) is a repulsive potential and @(R) therefore decreases with decreasing R. On the other hand, W(R) strongly decreases with increasing R, largely due to the decreasing barrier for proton tunnelling, since the other factors in eqn (4) are weakly dependent functions of R, and since we have ignored the electronic factor. The integrand of eqn (1) therefore has a sharp maximum at some value R* which depends t Strictly, within the partially adiabatic approach the electrons and the proton constitute the total quantum system.In semiclassical terms this means that I V(R)I 2Su,w(R) in eqn (2) is replaced by (AEu,w/2)2, where AEu,w is the splitting of the proton levels v and w. Since this quantity is approximately proportional to both Su,w and fin, our conclusions about the ApK and temperature dependence of the isotope effect are not affected, whereas the absolute values may be affected by a constant factor. As noted previously,l~208 THEORY OF PROTON TRANSFER REACTIONS on the concrete form of the proton potential (e.g., harmonic or Morse) and the repulsive potential U(R). We can thus replace eqn (1) by the equation 1* 2* where AR is the R-interval which effectively contributes to k. k = O(R*) W(R*) AR (5) R* is determined approximately by the equation pU'(R*) = [ W(R*)]-l d W(R)/dRR = n*.(6) of harmonic proton potentials for the ground levels only, i.e., So,o - - In order to see the physical meaning of this equation more clearly, we shall insert a parabolic repulsive potential of the form U(R) = +y(R- I)' and the Franck-Condon factor exp (- aR2).13 y characterizes the curvature of the repulsion potential and I its range, and a is a parameter which depends on the proton vibration frequencies in the initial and final states and on possible mixing coefficients ;12* 13* a is thus different for proton and deuteron transfer, i.e., a, # aH. Eqn (6) then takes the form Eqn (6) and (7) have the following implications : (a) for a sufficiently rapidly rising repulsion potential (with decreasing R) R* is independent of the characteristics of the tunnel barrier.In the analysis of kinetic isotope effects this implies in particular that the proton and heavier isotopes are transferred over the same distance when they tunnel between the ground vibrational levels.12* l 3 (b) When the potential does not rise sufficiently rapidly, i.e., when a/y is not much smaller than unity, the proton and its heavier isotopes tunnel over different distances. Since a,, > aH, the distance is smaller the heavier the isotope.12* l 3 ( c ) One implication of (b) is that the pre- exponential factor in the Arrhenius rate expression is smaller for the proton than for the heavier isotopes and that the activation energy is smaller due to a smaller repu1sion.l29 l3 However, we shall see below that for the majority of experimental results where such effects are observed, a slight thermal excitation of the heavier isotope, in contrast to the proton, provides a more convenient rationalization of the data.We conclude this section by introducing the Franck-Condon factors for the transferring proton, necessary for our subsequent quantitative analysis of experi- mental data. In general the proton transfer may proceed along a three-dimensional path involving mixing of modes.' 9 9 2o Although in principle this can be incorporated in the theoretical analysis, we shall take the simpler approach of representing the proton transfer by motion along a single mode only. However, we shall not restrict the description of this mode to the harmonic approximation,' '-l4 which is not expected adequately to represent the substantial proton transfer distances in question.Thus, we shall represent the proton or heavier isotope motion by Morse potentials, where the final state potential curve is inverted relative to the initial state curve, or by the symmetric squared hyperbolic tangent potential. These two potentials are more realistic representations of atom group motion along a stretching and a bending mode, respectively. Moreover, we shall incorporate, although only numerically, the possibility of large frequency shifts corresponding to the situation where the proton leaves the donor along a stretching mode and enters the acceptor along a bending mode or vice versa. The Franck-Condon factors of harmonic oscillators undergoing both coordinate and frequency shifts have been reported on several occasions.1* 2 - In the present work we have found it most convenient to calculate these factors directly by numerical integration of products of harmonic oscillator wavefunctions of the form R* x 1/(1+2a/y). (7) 4[f(Qi,,> = ( Z 3 / 2 j .0% ~ X P ( - Q&)Hj(Qi,f) (8)N . BRUNICHE-OLSEN AND J. ULSTRUP 209 j = v, w ; Qi = Q, Qf = Q-A, where A is the equilibrium coordinate shift, Hj(QiJ is the Hermite polynomial of degreej, and Q denotes the normal coordinate associated with the proton stretching or bending mode. On the other hand, for Morse potentials in the initial [h(Q)] and final [ff(Q)] state of the form .fi(Q) = D[1 -exp (-aQ)I2 ( 9 4 f X Q > = D i 1 - e ~ ~ [a(Q-A)l>’ ( 9 4 where D is the dissociation energy and a the anharmonicity constant [a = (hG/2D)+, where R is the proton frequency], the Franck-Condon factor is l5 X (P - 2 4 ( P - 2W)(P + r ( u + i)r(P + 1 - v)r(w + i)r(p + 1 - w) Su,w = ( ~ + 1 ) - “ + ~ ’ exp {-aA[p-(1+k)l/2)KI,_,,C(p+1) exp(-aA/2)] .(10) We shall apply the limiting r Kv(z) is the modified Bessel function of the third kind.21 asymptotic form valid for large argument, i.e.,21 and furthermore keep only the first term in this expansion. The corresponding vibrational energy levels are 2 2 E~ = Qh(j++)-+hQa2(j++)’. (1 2) It was shown previously l 5 that squared hyperbolic tangent potentials of the form 2 3 L(Q> = D th2 (aQ) h<Q> = Dth2 [a(Q-NI (1 3 4 (1 3 4 where the symbols have the same meaning as before, adequately incorporate the main features expected for atom group transfer along bending modes.In this case the Franck-Condon factors l5 and energy levels l 5 9 23 take the form F[+y-w+l, +p-$v-+w, p-v-w+k+Z; l-exp(2aA)I where B is the beta function, F the hypergeometric function 24 and respectively . c j = tiQ(j+$)-$kQa2(j++)’ -#iQa2,210 THEORY OF PROTON TRANSFER REACTIONS We notice finally that eqn (10) and (14) are valid when the proton transfer mode undergoes equilibrium coordinate shift only. If, in addition, frequency shift occurs, the Franck-Condon factors are calculated by direct numerical integration using wavefunctions of the form 229 23 4bf = N j exp (zi,f)z!pf-2j)/2Lp-2Ti(z P - J i,f ) where L is the Laguerre polynomial, zi = (p+l)exp(-aQ); zf = (p+l)exp[-a(A-Q)] Nj = [a(p - 2j)/r( j + l)r(p + 1 - j)]* ; p = 2a-2 - 1 and F [ - j , p + 1 - j , $ p + 1 - j ; +( 1 + th Zi.31 for Morse and squared hyperbolic tangent potentials, respectively.3. RESULTS OF MODEL CALCULATIONS We have performed numerical calculations of both the absolute values of kinetic isotope effects for proton and deuteron transfer reactions, { = kH/kD, and the dependence of on several important parameters, in particular A E and T. The isotope effect primarily arises from differences in Sv,w, EY and ~1." in eqn (4) when the proton is substituted by heavier isotopes. Thus, E&>, R& > E&), whereas SEW > SEW. In addition, the value of the remaining parameters, E,, AE and coeff, in principle undergo small changes when the isotope is substituted.Thus, first, the shift of AE = pK,--pK, generally amounts to 0.02-0.04 eV (ix., the acid containing the lighter isotope is the stronger) when the proton is substituted by a deuteron. Secondly, the long-range (coulomb) contribution to E, depends largely on the proton transfer distance and the geometry of the collision complex. If the deuteron transfer distance is smaller than the proton transfer distance (cf. section 2) the corresponding value of E, is therefore also smaller. Finally, both the short-range part of E,, weff, and the free energy of formation of the collision complex from the separated reactants [not included in eqn (4)] all depend on a variety of different molecular interactions and vibrational modes such as hindered translation and rotation of medium molecules as a whole, deformations of hydrogen bonds etc., which are also in principle isotope dependent.However, these effects are expected to be small; their absolute values are largely unknown. We shall, therefore, consider only the dominating effects of the isotope substitution, i.e., the effects reflected in the vibrational energy and Franck- Condon nuclear overlap factors associated with the proton transfer modes. We have chosen various values for E, and the proton transfer distance, AR, in the range 0.1-1 eV and 0.3-0.6 A, respectively ; in general the transfer distance is taken to be the same for all isotopes (implying that AH < AD), i.e., the repulsion potential is assumed to be a sufficiently strongly varying function of R. In a few cases (see section 4) this assumption has to be relaxed in order to obtain quantatitive agreement with experimental data.Proton stretching and bending frequencies are typical values of these modes known from spectroscopic data,g* 2 5 and the corres- ponding deuteron frequencies were obtained by multiplying these values by 2-*. Fig. 1-4 show the dependence of 5 on A E for various values of the important parameters E,, A and f2 for harmonic proton potentials. A maximum is always observed, in agreement with both predictions of classical theories of proton transfer and many experimental observation^.^ Moreover, the maximum is located at AE = 0N . BRUNICHE-OLSEN AND J . ULSTRUP 21 1 Y m Ii T I A i- I I -l.-- -1.0 -0.5 0.G 0.5 1.0 1.5 I AEIeV FIG. l.-tfplotted against A E for various values of Es.(a) ! 2 ~ = 3000 cm-I and AR = 0.37 8, (transfer along stretching modes), (b) ! 2 ~ = 1500 cm-' and AR = 0.62 A (transfer along bending modes), and the AE-scale of the latter is shifted by 3.0 eV. Values of Es are 0.25, 0.50, 1.0, 1.5 and 2.0 eV, and the larger &, the broader the curves. Harmonic potentials. I I I I I I . , I , -1.2 -a.a -0.4 0.0 0.4 0.8 1.2 - AE/eV FIG. 2 . 4 plotted against A E for various frequency values. l& = 1 eV, AR = 0.50A. Harmonic potentials. Roman numerals refer to the following frequency values : I : 1500 ; I1 : 2OOO; 111: 2500; IV: 3000; V: 3500cm-'.212 THEORY OF PROTON TRANSFER REACTIONS 56 La LO 32 n 3 2L * 16 a 0.C J I I I I I I * .2 -0.8 -0.4 0.0 0.4 0.8 1.2 AE/eV FIG. 3.--Kinetic isotope effect, 5, plotted against A E for various AR.Transfer along stretching modes (a, = 3000cm-'), and Es = 1 eV. Roman numerals I-V correspond to AR = 0.27, 0.32, 0.37, 0.42 and 0.47 A, respectively. Harmonic potentials. 15 12 n -Y % 9 Y K m II I -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0 8 1.0 AEIeV FIG. 4 . 4 plotted against A E for transfer along a stretching mode involving frequency shift. Es = l.OeV, AR = 0.37A. Roman numerals correspond to the following couples of frequency values (cm-') in the initial and final states (G, QL) : I : (2600, 3000) ; I1 : (2800,3000) ; I11 : (3000, 3000) ; IV : (3200, 3000) ; V : (3400, 3000). Harmonic potentials.N. BRUNICHE-OLSEN AND J . ULSTRUP 213 when the proton transfer is not accompanied by frequency shifts, whereas it is shifted to finite values of AE when such shifts do occur.This effect has also been observed e~perimentally.~~ 26 We notice furthermore the following features of fig. 1-4 : (a) c increases approximately exponentially with A2 as expected. For E, = 1 eV and Q = 3000cm-l the absolute value thus increases from 4 to 56, i.e., the range of most experimentally observed values, when the proton transfer distance increases from 0.27 to 0.47A (fig. 1). (b) The value of AR for " typical " c-values at the maximum (z 10) is x0.35 A. In view of the fact that this would be the proton transfer distance between the strongly hydrogen-bonded donor and acceptor water molecules in ice 27 this value seems low. On the other hand, this result is changed to larger values when Morse potentials are applied.(c) 5 also increases approximately exponentially, as expected, with increasing frequency. Such considerations are appropriate for proton frequencies in the range 1500-3000 cm-l, this being the n s X -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 AEIeV FIG. 5.-Plots of 5 against AE showing the effect of anharmonicity. E, = 1.0 eV. Curves to the left have Morse proton potentials, AR = 0.41 8, and D = 1, 4 and 7 eV with I lower the lower the value of D. Curves to the right (shifted 3.0 eV along the AEaxis) have squared hyperbolic tangent proton potentials, AR = 0.65 8,, QH = 1500 cm-l, and the same D values. The top curve in both families corresponds to harmonic potentials. approximate range of frequencies for stretching and bending modes along which the proton may be transferred.A consequence of this effect is that for a given value of 5, the value of AR is higher the lower the value of Q. Proton transfer along bending modes therefore proceeds over longer distances than for stretching modes (fig. 3). (d) The absolute value of 5 is largely determined by A and R, whereas the width of the maximum depends strongly on E,. This provides a possibility of obtaining the value of this important parameter in cases where it cannot be calculated from the curvature of the Brarnsted plot. In addition we notice that for sufficiently small values of EJhR > (2ESkT)%] the plot of 5 against AE displays an oscillating structure analogous to the one predicted both in free energy relationships for elementary The maximum is shifted to positive or negative values of AE when the process is accompanied by a decrease or increase, respectively, of the proton frequency.Fig. 5 shows similar plots when the Morse and squared hyperbolic tangent potentials of anharmonicity constants in the range 0.1-0.4 (dissociation energies in = 3000 cm-'. chemical processes and in vibrational structure of optical transition^.^' (4214 THEORY OF PROTON TRANSFER REACTIONS the range 1-7 eV) are applied. For the latter potential we have only shown data for a typical bending frequency, and for comparison the results for harmonic potentials are also shown. We notice that c decreases strongly with increasing anharmonicity for both potentials or the proton transfer distance resulting in a given value of c, increases.This is due to the smaller proton tunnelling barrier for the anharmonic potentials compared with the harmonic ones. Thus, for proton transfer along a stretching mode corresponding to a maximum isotope effect of 12 the proton transfer distance is 0.39, 0.40 and 0.46 A for a = 0.16, 0.22 and 0.43, respectively, or D = 7, 4 and 1 eV, respectively. We finally notice that for both stretching and bending modes the apparent activation energy, EA = d In W/d/3, for deuteron transfer is higher than for proton transfer, whereas the ‘‘ pre-exponential factor ”, A , is typically lower. For maximum values around 10 (Es = 1 eV) the differences are : for stretching modes Ei-Ef = 0.3 kcal and A D / A H = 0.13, for bending modes : EZ-E: = 1.7 kcal and A D / A H = 0.69 (AR is 0.4 and 0.65, respectively and D = 4 eV).This effect is often observed experimentally.8* In our present formalism it is due to a thermal excitation of the deuteron mode, even though the dominating contribution to the sum of eqn (4) is still provided by the ground vibrational levels. As expected, the effect is larger for larger values of IAEl, for lower values of the frequency (e.g., corresponding to bending modes), and for increasing anharmonicity. The reason for the higher excitation in the latter potentials is that the distance between the energy levels of the “ barrier spectrum ” is lower for the anharmonic potentials than for harmonic potentia1s.l. 4. APPLICATION TO EXPERIMENTAL DATA We shall now apply the theoretical framework outlined above to several sets of experimental data on the relationship between the primary kinetic isotope effect and the two fundamental physical parameters which characterize this quantity, viz.the free energy of reaction and the temperature. Thus, distinct maxima in the plots of < against AE appearing approximately at AE = 0 have been observed for the deproto- nation of nitroalkanes 9-31 and phenylalkane~,~~* 33 the deprotonation of a-carbon hydrogen in carbonyl compounds,26* 34* 35 the bromination of 3-nitro-( +)-camphor,3s the dehydrochlorination of 2,2,2-trichloro- 1,l -bis-(p-chlorophenyl)ethane, 37 for the acid catalysed hydrolysis of substituted vinyl ethers 38 and for the base catalysed hydrogen exchange of a z ~ l e n e s . ~ ~ Studies of the temperature dependence of 5 have been reported for the ionization of nitroalkanes by pyridine b a ~ e s , ~ O - ~ ~ the ionization of 2-carbethoxycyclopentanone 44* 45 and other carbonyl several base- induced elimination reactions, * 47-50 the deprotonation of 4-nitrobenzylcyanide by ethoxide ions 51 and the deprotonation of 4-nitrophenylnitromethane by several nitrogen Throughout our analysis we shall take the view that the primary kinetic isotope effect can be identified with the observed effects.This is justified in view of the small magnitude and relatively weak free energy dependence of secondary and solvent isotope effects,g* 33 and the fact that the systems considered involve proton or deuteron transfer to or from a carbon atom which does not exchange rapidly with the solvent.FREE ENERGY DEPENDENCE OF ISOTOPE EFFECT In several cases experimental investigations of the Brlansted relationship for the same system as those considered for the isotope effect analysis have been reported.N. BRUNICHE-OLSEN AND J . ULSTRUP 21 5 This would offer a possibility of an independent estimate of the important parameter E8. Thus, when the proton is transferred along a stretching mode, eqn (3) and (4) take the simpler form k M SfARS(R*) exp [ - fiU(R*) - B(Es + AE)2/4E,] (18) 2n: for all practical purposes, i.e., the dominant contribution in the sum is provided by u = w = 0. E, can therefore be determined by a least squares analysis of the curvature of the experimental relationship between k and ApK, i.e., where A = In ( o , ~ ~ / ~ ~ S ) - B U ( R * ) - B E , / ~ , B = 1.15 and C = (2.3)2/4fiEs.This analysis is not valid for proton transfer along bending modes nor for deuteron transfer since, due to the smaller frequencies, in both cases excited vibrational levels in the initial or final state contribute significantly even for small values of jApKI. Moreover, the experimental Brarnsted relationship frequently displays either a substantial scatter or a curvature which is too small for an accurate determination of E,. In these cases we have to fit a value to the experimental data for C(AE) the width of which is determined primarily by E,. In k = A +BApK+ C(pK)2 (19) IONIZATION OF NITROALKANES Systematic studies of kinetic isotope effects for reactions involving proton transfer from several nitroalkanes to both 0- and N-bases have been reported by Bell and G ~ o d a l l , ~ ~ Bell and Cox 30 and by Dixon and B r ~ i c e .~ ~ Fig. 6 shows a combined plot of the results of the two former. < shows a maximum where, however, the ascending branch is largely determined by variations in ApK resulting from changes in the (water+dimethylsuIphoxide) solvent. Since this is also expected to give rise to changes in E,, less attention should be given to these points. On the other hand, the remaining part, including the maximum, shows a sufficiently small scatter that a theoretical plot can be fitted reasonably accurately, even for different substrates (nitroalkanes) and acceptor bases (water, OH-, acetate, chloroacetate and pyridines). An exception is the base 2,6-lutidine which has an isotope effect much larger than indicated by the plot, presumably due to the larger proton transfer distance.g* 29 Although a Brarnsted plot is a~ailable,~’. 30 it shows too much scatter and too little curvature for an estimate of E,.On the other hand, the “ width ’’ of the plot of fig. 6 suggests a value of 0.7 eV using the harmonic approximation for the proton and 0.85 eV using the Morse potential (D = 4 eV). Moreover, appropriate typical values for the C--H stretching and bending frequencies are 2900 and 1400 cm-’, respectively and 3300 and 1700 cm-l for the 0-H stretching and bending modes, re~pectively.~~ Using these values the experimental data, including the shift of the maximum to slightly negative values of AE, can be reproduced if we assume that the proton and deuteron are transferred either along a stretching or along a bending mode, and for a transfer distance, AR, which is independent of the isotope.The experimental data, however, do not allow a distinction between these two alternative modes of proton transfer. The resulting values of E, and AR summarized in table 1 illustrate our previous conclusion, i.e., the lower the frequency and the higher the anharmonicity, the larger is the proton transfer distance which fits a given value of 5. The data by Dixon and Bruice 31 referring to aliphatic ammine bases are too scattered to justify an analysis, although a maximum in the isotope effect at ApK x 0 can be recognized. However, the data of Keefe and Munderloh 32 for ionization of phenylnitromethane by a variety of 0- and N-bases and the data of216 THEORY OF PROTON TRANSFER REACTIONS Bordwell and Boyle for ionization of 1 -phenylnitroethane and several m-substituted derivatives by piperidine, diethylammine and piperazine are shown in fig.7 and 8, respectively. The former data show a clear maximum in the isotope effect for ApK x 0 and, although a fair amount of scatter is displayed, the bases which show the largest discrepancy from the general trend are those which are also structurally 9 n 8 . s. x 7 ' 6 - 5 - 4 - 3 - 2 - -tz t 11 to 9 - 8 - 7 - 6 - 5 - L - 3 - - - 1 ' 1 I 1 I 1 - -1.0 - 0.6 - 0.2 0.2 0.6 1.0 AEIeV FIG. 6.-Plot of k ~ / k ~ against A E for ionization of nitroalkanes. 0 Experimental data of ref. (29) ; (.D ref. (30) for different solvent compositions.The curve corresponds to Es = 0.80eV, AR = 0.39 A, and transfer along stretching modes for which 52L = 2900 cm-' and i2& = 3300 cm-', and D = 4eV. 12 i 0 0 0 I ' I L 1 - -0.8 -0.4 0.0 0.5 0.8 A E FIG. 7.-Plot of k ~ / k ~ against A E for ionization of phenylnitromethane. The points are the experimental data of ref. (32), and the line is the best theoretical fit (Es = 0.6 eV, AR = 0.62A, transfer along bending modes ; Oh = 1400 cm-' ; SZL = 1700 cm-').N. BRUNICHE-OLSEN AND J. ULSTRUP 217 1 11 - 10 - 9 - 8 - 4 7 - 3 6 - * * 5 - 4 - 3 - 2 - I I a - -1.0 -0.6 -0.2 0.2 0.6 1.0 AEIeV FIG. &-Plot of k ~ / k ~ against A E for ionization of substituted phenylethanes. Points are from ref. (33) and the line is the best theoretical fit (Es = 0.7 eV, AR = 0.61 A, transfer along bending modes ; QL = 1400 cm-I ; Q& = 1700 cm-l).TABLE 1 .-BEST PARAMETER VALUES FOR FITTING EQN (4), (B), (lo), (16) AND (17) TO VARIOUS SETS OF EXPERIMENTAL DATA stretch nitro alk. phenyl-n. meth. phenyl-n. eth. 3-nit-( + )-camph . Et-n.-acet- Na-pr-2-one-sulph. Me, Et-ac. acet. ._ . _ _ harm Morse AR1 AR2 G--H N-H 0-H EL AR Es 0.7 0.37 0.85 0.39 0.42 2900 3300 0.5 0.36 0.5 0.38 2900 2700 3100 0.26 0.34 0.26 0.36 2900 2700 0.5 0.36 0.5 0.39 2900 3000 3000 0.9 0.36 0.9 0.36 2900 3000 3000 0.9 0.36 0.9 0.36 2900 3000 3000 bend nitro alk. phenyl-n. meth. phenyl-n. eth. h i t - ( + )-camph. Et-n.-acet- Na-pr-2-one-sulph. Me, Et-ac. acet. harm th2 ES AR EB AR1 AR2 C-H N-H 0-H 1.0 0.61 1.0 0.63 0.67 1400 1 700 0.6 0.61 0.6 0.63 1400 1600 0.7 0.60 0.7 0.61 1400 1700 0.26 0.54 0.26 0.56 2000 1700 0.5 0.59 0.5 0.60 1400 1700 1700 0.9 0.59 0.9 0.60 1400 1700 1700 1.4 0.59 1.4 0.60 1400 1700 1700 Es in eV, AR in A, and C-H, N-H and 0-H refer to the appropriate frequencies in cm-'.Furthermore, " harm ", " Morse ", and " th2 " indicate that the corresponding proton potentials have been applied, " stretch " and " bend " that the proton is transferred along stretching and bending modes, respectively, and AR, and ARz that the dissociation energy is 4 and 1 eV, respectively.21 8 THEORY OF PROTON TRANSFER REACTIONS most dissimilar to the bulk of the substances. We should also notice that the ascending and descending branches (with increasing AE) are largely constituted by N- and 0-bases, respectively.Following the same procedure as before we find quite good agreement between the experimental and theoretical results when bending frequencies (shown in fig. 7) and hyperbolic tangent potentials represent the proton, whereas the agreement is substantially poorer when high-frequency stretching modes in the harmonic or Morse approximation are applied. The resulting parameter values are given in table 1. The data of Bordwell and Boyle 33 (fig. 8) show less scatter, which is most likely due to the much greater structural similarity of the members of the series considered. The results are compatible with proton transfer along both stretching and bending modes ; the appropriate parameters are collected in table 1. 7 - 6 - 5 - A - 3 - BROMINATION OF 3-NITRO-( +)-CAMPHOR In this reaction bromine is used as a scavenger for the intermediate in the mutarotation of the The rate of reaction is, therefore, independent of the bromine concentration and determined by the base catalysed formation of the intermediate from the initial endo form, i.e., a proton transfer reaction.The dependence of the kinetic isotope effect on AE for proton transfer to various 0-bases is shown in fig. 9. The best fit to the experimental data is obtained here for proton A t n * X * \ 2t------ -0.3 -0.1 0.1 0.3 AEIeV FIG. 9.-Plot of kH/kD against AE for bromination of 3-nitro-( +)-camphor. Points from ref. (36) and the line the best theoretical fit (Es = 0.26 eV, AR = 0.44 A, transfer along bending modes ; QL = 2000cm-'; SZL = 1700cm-l). transfer along a stretching mode giving the parameters shown in table 1.We notice that E, values determined from the Brarnsted plot [which can be constructed from the data in ref. (35)] and from the " width " of the plot of the isotope effect practically coincide. This agreement is satisfactory, since the former only displays a fairly small mount of scatter. However, the frequency change necessary for the reproduction of the experimental data is opposite to the direction observed in most other cases for proton transfer from C- to 0-donor and acceptor fragments. IONIZATION OF CARBONYL COMPOUNDS The AE-dependence of the isotope effect of the proton transfer reactions between the carbonyl derivatives ethylnitroa~etate,~~ 26 ethyl- and methyl-acetoacetates, 9* 26* 35N .BRUNICHE-OLSEN AND J . ULSTRUP 21 9 sodiumpropane-2-one sulphonate 9* 34 and several 0- and N-bases has also been reported (fig. 10). For the ionization of ethylnitroacetate E, can be determined independently from the Brransted relationship using the data of ref. (26). Thus, including all bases investigated (H,O, OH-, CH2ClC00-, 2- and 4-picoline, 2,6-lutidineY phenoxide and 2-chlorophenoxide) a value of E, = 0.60 eV is found. If water is excluded the value 0.46 eV is found, whereas E, cannot be determined if both H20 and OH- are excluded. If we use the data of Bell and Spencer,56 where six carboxyl acids and 10 9. 8 - 7 - ' 6 - s 2 5 - 4 - 3 - - 2 - AEIeV FIG. 10.-Plot of k ~ / k ~ against AE for ionization of carbonyl compounds. 0 ref. (26) on ethyl- nitroacetate ; 0 ref.(26) on ethyl- and methyl-nitroacetates ; @ ref. (34). The lines are the best theoretical fits using stretching frequencies. Es = 0.5, 0.6 and 0.9 eV. AR = 0.39 8, ; RL = 2900 cm-I ; Q& = 3000 cm-'. I 1 I I 1 c two pyridine bases were investigated we find that E, is 0.46 and 0.52 eV, respectively, when the two pyridine bases are included and when they are not. These values are all reasonably consistent but differ from the value of 2.3 eV estimated by Cohen and The experimental data can be reproduced almost equally well by Morse and hyperbolic tangent potentials for the proton, and the corresponding best para- meter values are given in table 1. Much less comprehensive data are available for the other two systems, and as shown in fig. 10, only the descending branch of the isotope effect dependence on AE is available.The figure shows the best calculated plots, and table 1 gives the corresponding parameter values. DECOMPOSITION OF DIAZO-COMPOUNDS Both the Brsnsted relation and the dependence of the kinetic isotope effect on AE for the acid catalysed decomposition of several diazo-compounds have been r e p ~ r t e d . ~ ~ - ~ O While the smooth Brransted plot 5 8 9 6o for the decomposition of diazoethylacetate catalysed by several trialkylammonium salts permits a fairly unambiguous determination of E,, only the descending part of the plot of the isotope effect i s available. Moreover, the pK of the substrate can only be estimated within the limits of -5 and -2.59 For these reasons we do not show a figure relating220 THEORY OF PROTON TRANSFER REACTIONS to this system.The best fit for proton transfer is obtained if we assume that the proton is transferred from a bending N-H mode (1400 cm-l) to a stretching C-H mode (2900 cm-I), and for AR = 0.74A. ACID CATALYSIS OF 3-HYDROGEN EXCHANGE I N INDOLES The reaction scheme for these processes involves a rate-determining proton transfer from the acid catalyst to the indole followed by liberation of the proton or deuteron from the same carbon atom.61 The isotope effect for the deuterium-tritium exchange is almost constant (CD-T z 2) for a series of substituted indoles and different 0-acid catalysts corresponding to a free energy interval 0.1 2 AE 2 -0.6 eV. These observations are compatible with the theory outlined, if E, = 1-1.5 eV, A R = 0.6 A and the proton is transferred along bending modes.If stretching frequencies are applied, a more distinct maximum than the one observed experimentally is obtained. TEMPERATURE EFFECTS Evidence for nuclear tunnelling in chemical processes also rises from the study of the temperature dependence of the kinetic isotope effect in proton transfer and is manifested in the effects listed above. According to the present theory these effects must be ascribed to thermal excitation of either deuteron modes or proton bending modes, since the proton stretching modes are already frozen at room temperature. TABLE 2.-APPARENT ACTIVATION ENERGIES AND PRE-EXPONENTIAL ARRHENIUS FACTORS FOR DIFFERENT VALUES OF Es, AR AND PROTON VIBRATION FREQUENCIES QR = 1500 cm-1; D = 4 eV; AE = 0 Es 4.6 11.5 23 4.6 11.5 23 4.6 11.5 23 AR 0.58 0.68 0.75 AH/AD 0.80 0.73 0.76 0.56 0.57 0.61 0.42 0.45 0.50 AH/AT 0.40 0.39 0.38 0.19 0.22 0.22 0.11 0.14 0.14 EAH 1.23 3.19 6.19 1.46 3.46 6.48 1.66 3.69 6.72 A&-H 1.44 1.36 1.35 2.07 1.92 1.88 2.56 2.37 2.32 AET-H 2.56 2.42 2.36 3.57 3.36 3.29 4.36 4.12 4.03 DH = 3000cm-1; D = 4 e V ; A E - 0 ES 4.6 11.5 23 4.6 11.5 23 4.6 11.5 23 AR 0.35 0.42 0.52 AH/AD 6.9 5.7 4.9 13.8 10.5 8.2 47 23 20 AH/AT 20.8 12.3 9.2 49 25 17 151 62 38 EAH 0.86 2.6 5.5 0.86 2.6 5.5 0.86 2.6 5.5 AED-H 0.04 0.1 0.2 0.08 0.2 0.3 0.2 0.4 0.6 AET-H 0.3 0.5 0.7 0.5 0.9 1.0 1.3 1.7 1.8 Anharmonic potentials and a dissociation energy of 4 eV.Energy quantities in kcal, AR in A. In most cases the quantum effects are, however, weakly manifested in the tempera- ture dependence of EH-D or <.Thus, the appropriate activation energy difference, E:--EF, usually amounts to less than a single vibration quantum.s* Such effects are expected from the present theory, as seen from table 2 which shows the apparent activation energies and pre-exponential factors for proton and deuteron transfer reactions for representative values of E, and AR. The largest differences in activationN. BRUNICHE-OLSEN AND J . ULSTRUP 22 1 energy are clearly expected when transfer along bending modes occurs. If this is not plausible from the molecular structure of the reactants the possibility of different transfer distances for the proton and the heavier isotopes must be invoked [cJ eqn (7) and below].In the following we shall, therefore, only consider a few cases which either display " abnormally " large effects 5 8 or for which additional checks of the calculated parameter values can be obtained, by comparison between rate parameters for transfer of both proton, deuteron and triton. With reference to these criteria we shall thus, in turn, consider (Q) the ionization of 2-~arbethoxycyclopentanone,~~~ 45 (b) the ionization of nitroalkanes 40-43 and (c) the proton and deuteron transfer from 4-nitrophenylnitromethane to different N - b a s e ~ . ~ ~ ' ~ I ON1 Z AT1 ON OF 2-C A RBE THOXY C Y C LOP ENTA NON E The proton, deuteron and triton transfer between this substrate and D20, CH,C1000- and F- show both large isotope effects and considerable activation energy differences.Thus, table 3 gives the experimental values 4 4 9 45 for both the activation energies, EA, and the Arrhenius pre-exponential factor, A , for the various isotopes. Following the previous procedure, using literature values for ApK44g 45 and the appropriate vibration frequencie~,~ and assuming transfer along stretching modes, the best fit to the experimental data for the reaction with CH,ClC00- is obtained for Es = I .O eV, and AR = ARH = ARD = ART = 0.3 which give Ez = 11.0 kcal, AED-H = 0.3 kcal, AET-H = 0.7 kcal, kH/kD = 3.8 and kH/kT = 9.1, i.e., no satisfactory agreement for the activation energies. Fitting E, and AR to the latter gives isotope effects which are much larger than the experimental values. TABLE 3.-EXPERIMENTAL DATA ON ABSOLUTE VALUES OF k ~ / k ~ AND k ~ / k ~ AND ACTIVATION PARAMETERS FOR VARIOUS ISOTOPE TRANSFER REACTIONS system logAH E? logAD EZ 1ogAT E;f AED-H ~ O ~ A D I A H AET-a: I o g A r l A ~ k d k D k d k r 1 3.94 11.9 4.25 13.0 4.89 14.3 1.1 0.36 2.1 0.45 3.4 2 6.70 11.0 7.12 12.4 7.83 13.9 1.4 0.46 3.2 0.93 3.7 11.4 3 9.24 14.6 10.61 17.0 10.65 17.2 2.4 1.38 2.8 1.41 2.7 3.4 4 14.4 17.4 3.0 0.83 24 79 5 6.6 4.0 9.0 9.4 5.4 1.94 33 The system numbers 1-5 refer to the deprotonation of 2-carbethoxycyclopentanone by DzO, CH2C1C00- and F-, the deprotonation of 2-nitropropane by 2,4,6-trimethylpyridine and the deprotonation of 4-nitrophenylnitromethane by tetramethylguanidine, respectively.The activation energies are given in kcal mol-l. Application of typical bending frequencies gives similarly E z = 11.0 kcal, AED-H = 0.7 kcal, AET-H = 1.3 kcal, kH/kD = 3.8 and kH/kT = 8.6 for E, = 0.6 eV and AR = 0.59 A.When F- is the acceptor, the values E f = 14.6 kcal, AED-M = 0.1 kcal, AET-H = 0.3 kcal, kH/kD = 2.7 and kH/kT = 5.5 for E, = 1.5 eV and AR = 0.28 A and transfer along stretching modes, whereas Ef = 14.6 kcal, AED-H = 0.3 kcal, AET-H = 0.8 kcal, kH/kD = 2.67 and kH/kT = 6.0 are found for Es = 1 .OO eV, AR = 0.58 A and transfer from a bending to a stretching mode. Although the consistency of the theoretical data for bending modes is not bad and could be improved by inclusion of the isotope effect in some of the parameters where it has been ignored so far (e.g. the reaction volume or secondary isotope effects), we believe that the major cause of the discrepancy is the fact that the repulsion potential for the proton at small distances does not rise sufficiently rapidly with222 THEORY OF PROTON TRANSFER REACTIONS decreasing distance to ensure the effective equality of ARH, ARD and ART.Part of the differences in activation energy for kH, kD and kT therefore originates from the different repulsion energy, and the fact that ARH > ARD > ART [cf. eqn (7)]. In order to obtain an estimate of this effect we adopt the view that the repulsion potential can be satisfactorily represented empirically by a sufficiently simple function, e.g., harmonic or exponential and that the experimental activation energy for the proton transfer, E t , approximately represents contributions other than repulsion [the latter assumption can be relaxed if an estimate of U(R*) for the proton is avail- able].The difference in repulsion energy for the deuteron, AUD, then gives a contribution exp (AUD/kT) to the isotope effect. Using AUD as a parameter we can find the contribution ttun = to 5 arising solely from tunnelling, and from the activation energy of the deuteron transfer, (E2)tun = AUD, Stun and E z we can determine ARH, ARD and E,. Subsequently we can repeat this procedure for the triton transfer. If we ignore the small isotope effects on E,, however, it is more appropriate to use the value of E, already estimated. Together with the experimental values of AET-H and kH/kT this allows a determination of ART and AUT. Following the latter procedure we can reproduce the experimental data for the proton transfer to chloroacetate by the following set of parameters : E, = 0.9 eV, 0.9 kcal mol-' ; ART = 0.34 A, AUT = 2.2 kcal mol-l, i.e., we observe a monotonous dependence of the parameter values on the isotope mass as expected.When fluoride is the acceptor, the following values are found : E, = 1.6 eV, f2c--H = 2900 cm-', CH+ = 3450 cm-l ; ARH = 0.5 A ; ARD = 0.35 A, AUD = 2.3 kcal mol-' ; ART = 0.33 A, AUT = 2.2 kcal mol-l. The approximate equality between ARD and ART can be partly associated here with the relatively small difference between the reduced masses of DF and TF. C2c-H = 2900 cm-l; C2-H = 2700 cm-l; ARH = 0.49 A ; ARD = 0.41 A, AUD = IONIZATION OF NITROALKANES The proton transfer reaction between 2-nitropropane and the sterically hindered 2,4,6-trimethylpyridine shows both an " abnormally " large isotope effect, 9* an activation energy for this quantity which is substantially higher than the vibrational zero-point energy difference 41* 42 (see table 3), and a divergence from the general trend in the free energy relationship.Again using literature values for ApK29* 41. 42 and the vibration frequencies, the values of E, = 2-2.3 eV, ARH = ARD = ART = 0.77 A and typical bending frequencies can reproduce the experimental values of BE, kH/kD and AD/AH. This gives, however, kH/kT = 125, i.e., higher than the experimental value. Using stretching modes, the agreement is poorer, viz. E z = 11.3 kcal, AED-H = 0.7 kcal, kH/kD = 24 and kH/kT = 195 for E, = 2.0 eV and AR = 0.48 A. On the other hand, if we ascribe this discrepancy to the same effects as for the ionization of 2-carbethoxycyclopentanone, all the experimental data can be reproduced, if we choose E, = 2.4 eV, C2c--H = 2900 cm-l, = 2700 cm-l ; ARH = 0.7 A ; ARD = 0.57 A, AUD = 2.1 kcal rnol-l ; ART = 0.54A, AUT = 2.2 kcal mol-l.However, since no experimental data on E l are available, the choice of the last two parameter values cannot be made unambiguously. I ON1 Z A T I 0 N OF 4-NITRO PHENY LNI TROMET HAN E The proton and deuteron transfer from this compound to tetramethylguanidine, N,N'-diethylbenzamidine, tributylammine and triethylammine in several different solvents has been extensively studied recently. 52-5 These reactions generally showN . BRUNICHE-OLSEN AND J . ULSTRUP 223 large isotope effects ( k H / k D = 11-50), AI5D-H (1.5-5.4 kcal), A D / A H (1-30) and small AH and AD (log AH z 5-7, log AD x 6-91? which are all strongly indicative of proton and deuteron tunnelling.The most pronounced effect is observed when tetramethyl- guanidine is the base and cyclohexene the solvent; for this reason the appropriate data for this reaction are shown in table 3. Table 2 shows that effects of this magnitude cannot easily be associated solely with proton and deuteron transfer over fixed transfer distances. On the other hand, following the procedure outlined above the following parameter values are compatible with the experimental data (table 3) for the 4-nitrophenyl- nitromethane-tetramethylguanidine reactions : E, = 0.9 eV, Qzc-H = 2900 cm-I, = 2700 cm-l, D = 4 eV; A R H = 0.69 A; ARD = 0.53 A, A U D = 3.56 kcal mol-l.We also notice the relatively large value of ARH for both this and the previous reaction compared with most of the data discussed above. We notice finally that very large isotope effects (kH/kD > lo3) and an activation energy, Bz, which decreases strongly with decreasing temperature in the interval 77-120K has been observed for hydrogen and deuterium atom transfer from deuterated methanol and from acetonitrile, respectively, to methyl radicals in solid glasses.62* 63 While the former effect is compatible with the theory outlined above, the latter requires a modification of the formalism to incorporate the increasing quantization of medium modes with decreasing temperature at the low temperatures in question.Such an analysis is possible provided that the frequency spectrum of the medium is l6 However, since at least one additional parameter would then be introduced? and since the temperature interval investigated is not sufficiently wide to distinguish the role of the medium and the intramolecular modes, we shall not perform such an analysis here. 5. DISCUSSION We have shown that the quantum theory of elementary processes in condensed media provides an adequate description of all the important features of deuterium and tritium isotope effects of proton transfer reactions in homogeneous solution. This approach is fundamentally different from both the semiclassical approaches usually applied, which ascribe the isotope effect to loss of zero point energy when going from the initial to the transition state along the proton coordinate, and from the theory of Cohen and Marcus 63 according to which the isotope effect originates from differences in the ‘‘ intrinsic ” activation energies for the proton and deuteron transfer. The semiclassical approach cannot easily explain the qualitative dependence of t$ on AE and Tunless additional assumptions about the force constants of the transition state and the possibility of proton tunnelling are involved.In contrast, the formalism outlined and applied in the present work is based on the fundamental theoretical result that the proton and the solvent (including other classical modes) are repre- sented by separate (although possibly coupled) modes, and both sets exert important but different roles in the activation process. In addition, the rate expressions contain parameters of initial and final states only, i.e., information which is in principle experimentally available ; difficult estimates of transition state Parameters are thereby avoided.Finally, the theory represents an extension of previous applications incorporating anharmonicity effects for the proton potentials and by inclusion of all excited vibrational states in quantitative estimates of the various parameters. Our results can be summarized in the following way : of multiphonon electron and atom group transfer theory to isotope effects 12* l3 b Y224 THEORY OF PROTON TRANSFER REACTIONS (A) the presence of an isotope effect is ascribed to the fact that the proton and deuteron are generally transferred by tunnelling and that the tunnelling is easier the lighter the isotope.However, in most cases the barrier for tunnelling only contributes a small fraction of the activation energy. The dominant contribution to the fatter is provided by the solvent and other classical modes, i.e., nuclear motion along coordinates which do not refer to the proton. (B) For proton transfer along stretching modes and in cases where the Brarnsted coefficient is ~ 0 . 5 , only the ground vibrational level of the proton contributes significantly. For the deuteron a small but significant contribution is also provided by the first excited level, either in the initial or in the final state. This effect increases with increasing IApKl and is much more pronounced for transfer along bending modes, in which case excited states of the proton also contribute.Moreover, the effect is the fundamental cause of the dependence of on both AE and T. This conclusion differs from that of previous applications of the theory to kinetic isotope effects which ascribe the temperature dependence of 5 largely to different transfer distances for the proton and the deuteron and consequently to different values of the repulsion potential . (C) The experimental dependence of 5 on AE is always quantitatively reproduced by the theory for reasonable values of E, and AR = ARH = ARD. These two important parameters can furthermore be determined independently, from the Brarnsted relationship or the width of [(AE), and from the absolute value of 5 at the maximum, respectively.(D) The dependence of [ on AE is not strongly affected by possible differences in ARH and ARD, and in no case is it necessary to invoke such an assumption. (E) On the other hand, in a number of cases the experimental data on the dependence of 5 on T imply that AR decreases with increasing mass of the isotope, i.e., the repulsion potential between donor and acceptor varies less rapidly with R than the transmission coefficient. (F) Anharmonic proton potentials give smaller isotope effects or larger isotope transfer distances for a given value of the isotope effect compared with harmonic potentials. For Morse and squared hyperbolic tangent potentials the latter effect amounts to approximately 10 % for representative parameter values.However, the " resonance " splitting in the region of the reaction hypersurface may increase the " effective " anharmonicity of the proton potentials, and the effect may, therefore, be larger than estimated in the present work. We notice finally that transfer along stretching and bending modes can often equally well account for the observed effects. This means that a more involved path implying a mixing of initial and final state modes would also do so. This is also considered in semi-classical theories of proton transfer. In principle, such a path can be quantitatively incorporated in our calculations as well l2 when either the geometry or the total three-dimensional interaction potentials are known. Note added in proof: Recent calculations on the kinetic isotope effect, published after submission of the present work, has provided concIusions similar to ours concerning the dependence of k ~ / k ~ on ApK and Es.64 We would like to thank Prof.J. P. Dahl, Chemistry Department B, Lyngby, and Drs. E. D. German and A. M. Kuznetsov, Institute of Electrochemistry of the Academy of Sciences, Moscow, for helpful discussions. (a) R. R. Dogonadze and A. M. Kuznetsov, Physical Chemistry, Kinetics (VINITI, Moscow, 1973) ; (b) Progr. Surface Sci., 1975, 6, 1. N. R. Kestner, J. Logan and J. Jortner, J . Phys. Chem., 1974, 78, 2148.N. 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