Energetic and Dynamical Aspects of Proton Transfer Reactions in Solution BY R. A. MARCUS Department of Chemistry University of Illinois Urbana Illinois 61 801 USA Received 19th May 1975 Several energetic and dynamical aspects of proton transfers are treated. The effect of intrinsic barrier asymmetry on BEBO calculated Bronsted plots is investigated and contributions to work terms are also considered. The dynamics of transfer of a light particle between two heavier ones is discussed for a particular potential energy surface making use of classical trajectories semidassical concepts and a previous quantum study. The question of nonequilibrium polarization of solvent is also considered. 1 INTRODUCTION It is a pleasure to participate in this symposium honouring Professor R.P. Bell whose work has illuminated so many parts of the proton transfer field. In this paper I would like to comment on several aspects of proton transfer both energetic and dynamic (1) effect of '' intrinsic barrier asymmetry " on Bronsted plots (2) dynamics revealed by recent classical and quantum mechanical studies for an H-atom transfer (3) contributions to the "work terms " and (4) the possibility as in electron transfers of nonequilibrium polarization of the solvents. 2 INTRINSIC BARRIER ASYMMETRY AND BRONSTED SLOPES Sometime ago we considered a model of a proton transfer reaction,' AiH+A2 +Ai+HAZ (2.1) (charges are omitted for notational brevity) in which the process occurred in three steps AIH+Az + AIH-* *A2 (2.2) AIK--A2-+ Al.* *HA2 (2.3) Al** *HA2-+ Al +HA2. (2.4) Of these only the middle one depended on the standard free energy of reaction AGO' for (2.1).* Step (2.2) involves a free energy change w' (called a " work term ") for bringing the reactants close together ; wr includes steric (orientation) effects and where necessary,2 partial desolvation. The next step (2.3) is the actual proton transfer and involves intramolecular and solvent reorganization to form the transition state followed by an intramolecular and solvent relaxation. Step (2.4) is a " disorienting " and resolvating one ; it contains a work term -wp wp being the analogue of w' for the reverse reaction. * AGO' is actually the " standard " Gibbs free energy change in the prevailing medium and at the prevailing temperature.60 R. A. MARCUS In an approximation which was quadratic for treating (2.3) and which at the same time neglected “ ),-asymmetry ” (3.f-A2 defined later) the rate constant was given by k = Z exp( -AG*/kT) (2.5) apart from the usual statistical factors.‘k 2 is the collision frequency in solution AG* is AG* = ~‘+(R/4)(1+AGg’/i)~ (IACi’I < i) (2.6) and AG;’ = AG” +W” -w‘. (2.7) i is the “ intrinsic ” free energy barrier,’ i.e. the barrier in (2.3) when AG;’ = 0. AG;’ is seen from eqn (2.7) to be the effective standard free energy of reaction for the proton transfer step itself. Similarly in a bond energy-bond order (BEBO) type of calculation the corres- ponding vaIue of AG* is given by eqn (2.8) when i-asymmetry is neglected and when the E’s in ref.(1) are replaced by free energies AG* = w‘ +044) +(AG;;’/2)+()./4y)ln cosh(2yAG;;’/l.) (2.8) where y is ln2. The difference between eqn (2.6) and (2.8) was typically relatively small.’ Implica-tions of the equations are apparent a small 1implies a large curvature of a Bronsted plot; a small 2 also implies a large limiting rate at large negative AGi’ when w‘ is small but the limiting rate is small when wr is large. A question which arises is the effect of ).-asymmetry. Specifically if a potential energy surface is varied by varying AGO‘ of reaction (2.1) holding constant the intrinsic barriers of the exchange reactions AiH+A + A,+HA, (i = 1 2) (2.9) do the preceding considerations prevail ?3 The intrinsic barrier E.J4 for the reaction in eqn (2.9) may depend on i and the difference in is called here the 2.-asymmetry Differences in 2 and L2 were neglected in deriving eqn (2.6) prompted in part by a finding that such effects were relatively minor in the quadratic case i.e.in eqn (2.6).‘ We consider now their effect on the BEBO derived formula eqn (2.8). The problem is how to calculate the effect of varying AGO‘ holding the intrinsic barriers constant and not assuming I, = Typically a potential energy surface is not automatically characterized in terms of ,il 2 and AG;’. For example in a BEBO model for the reaction in eqn (2.9) the potential energy of formation of an intermediate state can be written as AE = V,-Vln?‘-V2n4* (2.10) where along the reaction path bond order is conserved 12’ +?I = 1 (2.1 I) n and Viare the bond order and bond energy of the A,H’th bond pi is an exponent which reflects a property of that bond.For the exchange reaction in eqn (2.9),ni is 5 in the transition state and so AE for that reaction which we may call AEl or better yet Ai/4 is found from eqn (2.10) to be Vi[I -2(+)p’]. * If srand sp are the statistical factors for forward and reverse steps. it suffices to replace wr and WP by Wr-kTIn sr,wP-W In sP to include their effect.’ Further k is the k in footnote 3 Of ref (1) in the case of diffusion effects. DYNAMICS OF PROTON TRANSFER The potential energy change accompanying the reaction in eqn (2.3) is A V AV = V1-V2. (2.12) Thus the effect of AV on the potential energy barrier to reaction A€ can be investi- gated holding the intrinsic barriers l1and E.constant only by varying p1 and/or p2 simultaneously. (It is not clear that this precaution was followed previ~usly.)~ The value of til in the transition state is obtained by setting dAE/dnl = 0 and intro- ducing the resulting it1 and n2 into eqn (2.10). Investigation of the effect of A V on A€ holding R and Az constant is considerably simplified as in eqn (10) of ref. (l) by noting that p z 1 and expanding nf' in eqn (2.10) in a Taylor series retaining only the first two terms. The barrier A€ is found (eqn (12) of ref. (1)) to be AE = n$AV-(&4y)tii In nf (2.13) i= 1.2 where it! and ni are the solution of dAE/dnf = 0 i.e. of 0 = -AV-(Al/4y)(ln nfi + 1)+(i2/4y)(h nf,+1) (2.14) nt+nt = 1.(Eqn (2.8) can be obtained from eqn (2.13) and (2.14) by setting ;Cl = L2 = I. replacing AV by AG;' and adding to (2.13) the barrier wrof the first step (2.2).) Now at last AE depends only on AG;' and 12. The slope of a (AE AV) plot at a given A1 and 11 is obtained by observing that dAE/dAV is the sum of (dAE/JAV,,?) and of (aAE/an?)dY(anf/aAV).Since (dAE/dn\)Av is zero one finds from eqn (2.13) that dAE/dA V = n!. (2.15) The A€ in eqn (2.13) can be obtained by first introducing values for n? and ni into eqn (2.14) solving the latter for AV and introducing this result into eqn (2.13). In table 1 the results of such a calculation are given choosing a rather large asym- metry A1/4 = 12 and A2/4= 2.TABLE 1.-EFFECTSOF REACTANT ASYMMETRY ON (AE,A V)PLOTS n2 AV AE nz' AV AE 0.1 -19.2 0.4 0.6 0.0 7.2 0.2 -15.2 1.0 0.7 5.4 10.7 0.3 -11.7 1.8 0.8 12.8 16.3 0.4 -8.2 3.1 0.9 25.1 26.9 0.5 -4.4 4.8 From table 1 one sees that the l./4 in eqn (2.6) namely AE at AV = 0 is 7.2. The latter is close to (Al +A,)/2. The Bronsted slope dA€/dAV for the system is seen from eqn (2.15) to be n$. Thus when the true slope is 0.1 0.3 0.6 and 0.8 say one finds from the above A and the corresponding AV's in table 1 that the slope calculated from eqn (2.6) is 0.17 0.30 0.50 and 0.72 respectively values which are fairly close to the true slopes. 3 SOMEDYNAMICAL ASPECTS OF LIGHT PARTICLE TRANSFER Chemical kinetics has received additional insight from recent studies with mole- cular beams lasers and infra-red chemiluminescence.6 On the theoretical side the main method for interpreting these data has involved computer-calculated classical R.A. MARCUS trajectories of the atoms,6b* because of the difficulty of solving the fully three- dimensional reactive collision problem numerically and quantum mechanically. Numerical quantum mechanical studies have been almost entirely confined to collinear collisions.* In the case of proton transfers no trajectory or quantum mechanical numerical studies appear to have been made as yet. Some insight into the dynamics can be obtained by studying instead the transfer of a hydrogen atom between two heavier particles. The only quantum mechanical study which has appeared is that of a collinear collision between HBr and C1.C1+ HBr +-ClH + Br using a London-Eyring-Polanyi-Sat0 potential energy surface. This limitation of collinearity is perhaps not in itself too dismaying; the actual collisions in solution with major steric or solvation features can differ substantially from the usual three- dimensional gas phase collisions. The transmission probability was calculated for the reaction and more specifically for the formation of various vibrational states of the product HC1 of this exothermic proce~s.~To analyze the results of this study and to obtain implications for other light particle transfer Dr. Ellis of this laboratory has undertaken some classical trajectory studies on this and related systems. While the results will be described elsewhere,lO some features are summarized below.3 4 R,a u FIG. 1.-Skewed-axes plot of potential energy contours for reaction (3.1). R1 = RCI-H R2 = RH-B~/C,where C is the usual mass-scaling factor l1 (0.987 here). The dotted line denotes a transition state and a reactive trajectory is also indicated. A diagram of the surface used is given in fig. 1 in the usual skewed-axes form? (As is well known plots in rectangular-axes form while frequently used are misleading for purposes of analyzing the dynamics of individual trajectories.) The radial coordinate is essentially a scaled C1- Br distance while the angular coordinate is the protonic coordinate. In one definition the transition state is the line of steepest ascent from the saddle-point indicated in fig.1 by the dotted line. The latter is seen to be curved in the present highly exothermic instance. A typical trajectory for reactants with an initial zero-point vibrational energy and with a substantial initial translational energy (9 kcal/mol above the barrier height of 1 kcal/mol) is indicated in fig. 1. For most of the trajectories corresponding to these and lower energies the relevant part of the dotted line is effectively perpendicular to the horizontal axis. Thereby the reaction coordinate in this appreciably exother- mic system is essentially the C1 -Br distance. We found that the classical probabilities agreed approximately with the quantum mechanical values for the transitions which were classically allowed i.e.those for which the final vibrational states of products were attainable from the initial ones of reactants via real-valued classical mechanical trajectories. (Classically-forbidden transitions are those which require complex-valued trajectories. 2 A substantial fraction of the trajectories which passed through the transition state region (i.e. across the dotted line) did or did not recross it to reform reactants depending on the initial translational energy. The behaviour in the preliminary DYNAMICS OF PROTON TRANSFER studies appears to suggest that a proper phasing of the H-and C1-Br motions is needed for reaction. The recrossing itself " wastes " phase space. It implies that apart from tunnelling corrections the rate will be typically less than that predicted by transition state the~ry.'~ However even a factor of three as a discrepancy between transition state theory and the actual dynamics is a minor one considering the large variations in rate which can be studied by variation of factors such as AG;'.A second deduction can be made from the classical trajectories using semi- classical l4 arguments Because the zero-point energy of the vibrational motion (more precisely a vibrational " action variable " J)is roughly constant up to the transition state region in the above study the vibrational motion is substantially " adiabatic " l4 in this region of space. The "quantum number " of the vibration N is related to J by the well-known Bohr-Sommerfeld eqn (3.2) for a vibrational coordinate a formula later justified by the WKB solution of the Schrodinger equation.J = (N+t)h. While N can have any real value classically but only integer values quantum mechani- cally the same approximate adiabatic behaviour which led to a tendency to preserve J classically in the present case in the region up to the transition state will lead to a similar tendency to preserve N quantum mechanically in that spatial region. The vibrational energy is for a harmonic oscillator of frequency v equal to Jv both classically and quantum mechanically. Thus apart from minor variations of v in this region the vibrational energy is also roughly constant. Since isotopic effects on the rate constant in the absence of tunnelling are largely attributed to differences in zero-point energies of reactants and the transition state,' there should be essentially no isotopic effect on the rate constant in this appreciably exothermic system when H is substituted for D.Finally a type of Franck-Condon principle also operates in the region where the system moves from one channel to another the momentum of the " slow " coordinate C1-Br being substantially conserved in that region. Here the protonic motion is very nonadiabatic and a significant increase of its vibrational action (and energy) occurs. Thus in the reverse reaction vibrational energy should facilitate the proton transfer an effect which might be observable in a suitably stabilized (e.g. intra- molecularly hydrogen-bonded) system using short laser pulses. In the case of the corresponding thermoneutral system Cl+HC13 ClH+Cl (3.3) the potential energy surface is quite different from that depicted in fig.1. The surface is now symmetrical about the bisector of the acute angle and the dotted line repre- senting the transition state now lies along that bisector. The reaction coordinate is in the vicinity of the transition state perpendicular (as before) to the dotted line and so now is substantially a motion of the proton. The original zero-point energy of the protonic motion has thus been lost or really converted to motion along the reaction coordinate when the system passes across the dotted line region. The full effect of an H and D isotopic difference in zero-point energy is thereby felt yielding a maximum isotope effect (tunnelling corrections aside).These facts are well-known,I5 but it is interesting to see them borne out by the behaviour of the trajectories. The various dynamical results classical and semiclassical thus have implications for approximate dynamical treatments of light particle transfer but we shall omit here further discussion of them. The above remarks apply to potential energy surfaces such as that in fig. 1 and its analogues for less (or more) exothermic reactions. In the case of proton transfers R. A. MARCUS in solution the effective surface is more apt to have potential energy wells in the two channels rather than free escape channels out to infinity wells created by hydrogen bonding or by cage effects. Nevertheless from semiclassical considerations effects similar to those described above are expected to apply in this case also.4 WORK TERMS wr AND wp The work term can be a composite of several terms. In the case of carbon acids or bases which do not participate in hydrogen bonding some desolvation of an attacking nitrogen oxygen base or acid may be needed and not compensated for by a favourable AG;' and so contribute a term wiesto wr. Again in the large molecules which are usually involved and when the reactants are not joined by hydrogen bonding an appreciable steric restriction may occur and contribute a term wit. For example in the gas phase abstraction of a hydrogen atom from an alkane by a methyl group CH3+HR + CH4+R (4.1) a steric factor of the order of can be anticipated,16 and would correspond to a work term wr of about 4 kcal/mol.Such steric factors might be reduced somewhat by favourable AGi' but only a slight effect would be anticipated in the present case. If one assigns to the partial desolvation a contribution of the order of 6 kcal/mol and assumes a steric effect of the above magnitude the net wr for nonhydrogen bonded reactants would be about 10 kcal/mol which is of the same order as that needed to explain the data.2* Another contribution to the work term can also occur when the immediate product of the third step in the reaction eqn (2.4),is not the separated products but rather is a metastable intermediate which later ruptures (cf. eqn (5.1) later). Whenever this last step has an activation barrier Wiec which exceeds the barrier for the intermediate to reform the reactants this wiec should in effect be added to the previously computed free energy barrier.We then have W' = Wies +Wft +wiec. (4.2) Of these w' contributions only the first two contribute to the w' in eqn (2.7). 5 NONEQUILIBRIUM SOLVENT POLARIZATION In electron transfer reactions a charge transfer occurs between two reactants and the " charge centres " are usually some 5 to 10 A apart. In the transition state the electron cannot be in both places at the same time and the solvent orientation- vibrational polarization adopts a value which is some compromise. The solvent electronic polarization on the other hand can largely follow the motion of the electron being transferred. This situation where the nuclear part of the solvent polarization is not that dictated by either charge centre alone and where the electronic part is dictated by the instantaneous position of the transferred electron and by the field due to the nuclear part was termed " nonequilibrium polarization " and treated in some In the case of a simple proton transfer between two adjacent centres as in eqn (2.3) the charge is transferred only over a relatively short distance and an effect such as the above would be expected to be minor.In some cases however the assumed mechanism involves rearrangement of several bonds with a somewhat larger dis- placement of charge in the proton transfer step (5.1). One example might be AH+R1RzC=N+=N-+ A-+RIR2CH-N+ N (5.1) (followed by elimination of N2and by other processes).S 10-3 66 DYNAMICS OF PROTON TRANSFER To obtain the potential energy of the transition state for any given configuration of the nuclei of the reaction complex and of the surrounding solvent the Schrodinger equation is solved for the electronic wave function. When attention is focused on the electrons of the reactants and the electrons of the solvent are treated for reason of simplicity as forming a polarizable dielectric continum one obtains a nonlinear Schrodinger equation. The free energy of formation of a nonequilibrium polarization state with an arbitrary orientation-vibration solvent polarization is given by W,, = -[(I -1/D0,)/8n] J D2dr-J P . D dr+2nc JP2dr (5.2) neglecting dielectric image effects.D(r)is the field directly due to the charges on the reactants l/c is l/Dop-l/Ds r is any point in the solvent P(r) is a function of the arbitrary orientation-vibration polarization and Do and D are the optical and static dielectric constants of the solvent respectively. Ultimately eqn (5.2) can be replaced by a more rigorous statistical mechanical expression but it will suffice for purposes of the present discuqsion. The Schrodinger equation for the wave function $ of the electrons of the reactants for any nuclear configuration Y of the reactants and (positions) of solvent molecules is obtained by minimizing 2o the following functional p($) with respect to ,$ at a given P. (5.3) where r1 denotes the totality of coordinates for the reactants’ electrons and I V$ I2 really denotes a summation over such electrons a b .. . I Va$ l2 + I Vb$ l2+ . . . ; V(r,r,) includes the potential energy arising from interactions within the reactants and with the solvent molecules apart from that included in the relatively long-range polarization term W,,,. UItimately all values of the r are considered and a suitable quantum and statistical mechanical average is made over r,. The D appearing in eqn (5.2) is 1/1 r-r1 I being an abbreviation for a sum over reactants’ electrons 1 /I r-ra I + 1/1 r-ryb I+ . . . When the resulting (nonlinear) Schrodinger equation is solved for $ one obtains a $ which depends on P(r). p($) then becomes a function of P which can then be obtained by then minimizing p with respect to P.In the case of electron transfer reactions it was possible to introduce a simplifying approximation writing $ as a linear contribution of two terms with weak overlap between them one term being the same as for the reactants and the other being the same as for the products and both reactants treated as spherical.18 The results obtained from eqn (5.2)-(5.4) can be shown (Appendix 1) to be equivalent to those obtained l8 earlier by a different and in some respects less general method. To the extent that the electronic wave function for the transition state of the reaction in eqn (5.1) could be similarly approximated for this purpose,21 the previous 4* * results for electron transfers could be adapted to that for proton transfer and added to the contribution to AE in eqn (2.10).When $ cannot be written as a linear combination eqn (5.2)-(5.4) remain applicable but more formidable. Elect-ronic structure calculations for the transition state of reactions such as (5.1) would therefore be helpful. R. A. MARCUS When the electronic energy of the system has been obtained as a function of r and P,the latter remain to be treated statistically as in transition state theory or dynamically. Examples of dynamical treatments for other or related potential energy surfaces are given in ref. (21) and (22). 6 SUMMARY A substantial " reactant asymmetry '' does not have a large effect on the slope of Bronsted plots (Section 2). Possible contributions to the work terms are summarized in Section 4 and the relation of the nonequilibrium polarization study in electron transfers to a possible one in proton transfer is considered in Section 5.On the dynamics side some results and implications of a recent study of dynamics of light- particle transfer are described in Section 3. APPENDIX 1 RELATION OF EQN (5.3) TO THOSE IN REF. (18) If $l denotes the electronic wave function for the pair of reactants as in ref. (18) and t,b2 denotes that for the products a trial $ is This $ is introduced into eqn (5.3) and the variation 69 is calculated at fixed P,and set equal to zero. The 6cl and 6c2 are subject to Cl+C2 = 1. (A2) When the assumption of weak overlap of t+bl and $2 is imposed one can show that one obtains the result that the free energy of reactants with an arbitrary P equals that of the products in this same P environment.This condition is identical with that imposed in ref. (18) to satisfy the Franck-Condon principle for these weak overlap systems. One next finds P by minimizing 9subject to this new constraint obtaining a relation the same as that used in ref. (18). The results in that paper are then obtained when the approximation of spherical reactants is introduced. This work was supported in part by the Office of Naval Research. R. A. Marcus J. Phys. Chem. 1968 72,891. M. M. Kreevoy and D. E. Konasewich Ah. Chem. Phys. 1971 21 243. G. W. Koeppl and A. J. Kresge J.C.S. Chem. Comm. 1973 371. R. A. Marcus J. Chem. Phys. 1965 43 679. H. S. Johnston Adv. Chem. Phys. 1960 3 131. E.g.(a)J. P. Toennies Physical Chemistry an Advanced Treatise ed. H. Eyring D. Henderson and W. Jost (Academic Press New York 1974) vol. 6A chap. 5 ; (b)articles in Faraday Disc Client. SOC. 1973 55 and ref. cited therein ; (c)T. Carrington MTP International Review of Science Physical Chemistry Series One Chemical Kinetics ed. J. C. Polanyi (Butterworths London 1972) vol. 9 p. 135 ; J. L. Kinsey p. 173. 'D. L. Bunker Molecular Beams and Reaction Kinetics ed. Ch. Schlier (Academic Press New York 1970) p. 355 ff; M. Karplus p. 372 ff; J. C. Polanyi and J. L. Schreiber Physical Chemistry an Advanced Treatise ed. H. Eyring D. Henderson and W. Jost (Academic Press New York 1974) vol. 6A chap. 6. E.g. E. M. Mortensen J. Chem. Phys. 1968 48,4029; D. G. Truhlar A. Kuppermann and J.T. Adams J. Chem. Phys. 1973 59 395 and refs. cited therein ; for 3-D calculations see A. Kuppermann and G. C. Schatz J. Chem. Phys. 1975 62 2502 and A. B. Elkowitz and R. E. Wyatt J. Chem. Phys. 62 2504. M. Baer J. Chem. Phys. 1972 62 305. lo R. L. Ellis and R. A. Marcus to be published. S. Glasstone K. J. Laidler and H. Eyring The Theory of Rate Processes (McGraw-Hill New York 1941). DYNAMICS OF PROTON TRANSFER l2 J. Stine and R. A. Marcus Chem. Phys. Letters 1972,15,536 ; T. F. George and W. H. Miller J. Chem. Phys. 1972,56 5668. l3 E. Wigner Trans. Faraday SOC. 1938 34,29. l4 Cf. R. A. Marcus Techniques of Chemistry Investigation of Rates and Mechanisms of Reactions ed. E. S. Lewis (John Wiley and Sons New York 1974) vol.6 pt. 1 chap. 2 ; ref. (50) and (51) cited therein. l5 R. P. Bell The Proton in Chemistry (Cornell University Press Ithaca New York 2nd ed. 1973) ; R. P. Bell Chem. SOC. Rev. 1974 3 51 3. l6 E.g. J. A. Kerr Free Radicals ed. J. K. Kochi (John Wiley and Sons New York 1973) vol. 1 chap. 1 p. 15. A. J. Kresge S. G. Mylonakis Y. Sato and V. P. Vitullo J. Amer. Chem. SOC. 1971 93 6181 ; W. J. Albery A. N. Campbell-Crawford and J. S. Curran J.C.S. Perkin 11 1972 2206; M. M. Kreevoy and S. Oh J. Amer. Chem. SOC. 1973,95,4805 ; A. J. Kresge Chem. SOC. Rev. 1973 2,475. R. A. Marcus J. Chem. Phys. 1956 24,966 979. l9 R. A. Marcus J. Chem. Phys. 1965 43 3477 Appendix I. 2o Cf. S. I. Pekar Untersuchurlgen iiber die Elektronentheorie der Kristalle (Akademie Verlag Berlin 1954).21 For a weak overlap approach see also V. G. Levich R. R. Dogonadze and A. M. Kuznetsov Electrochim. Acta 1968 13 1025 ; Electrokhym 1967 3 739. 22 E.g. K. D. Godzik and A. Blumen Phys. stat. sol. 1974 66B 569 and ref. cited therein ; S. F. Fischer G. L. Hofacker and M. A. Ratner J. Chem. Phys. 1970 52 1934; S. F. Fischer and G. L. Hofacker Internat. Symp. Phys. of Ice Munich 1968 eds. N. Riehl B. Bullemer and H. Engelhardt (Plenum Press New York 1969) p. 369.