J. Phys. Chem. 1986,90, 3166-3774
3766
removed from a singly charged ion in the forward direction and from a doubly charged ion in the reverse direction. In each case, therefore, the reverse reaction will involve desolvation of the more heavily charged species, which will require more energy. It is thus not unreasonable that wp should be larger than w'.
Acknowledgment. We are grateful to the Natural Sciences and Engineering Research Council of Canada and the donors of the
Petroleum Fund, administered by the American Chemical Society, for their financial support of this research, Registry No. Isobutyrophenone enol, 4383-10-2.
Supplementary Material Available: Tables Sl-S6 of rate data for the enolization of isobutyrophenone and the ketonization of isobutyrophenone enol in aqueous solution at 25 OC (26 pages). Ordering information is given on any current masthead page.
Phenomenological Manifestations of Large-Curvature Tunneling in Hydride-Transfer Reactions Maurice M. Kreevoy,* Draien Ostovit, Donald G. Truhlar,* Chemical Dynamics Laboratory, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455
and Bruce C. Garrett Chemical Dynamics Corporation, Columbus, Ohio 43220 (Received: February 7, 1986)
An important consequence of recent dynamical theories of tunneling is that, because of large curvature of the reaction path in a typical H, H ' , or H-transfer, light-isotope transfer occurs in more extended nuclear frameworks than heavy-isotope transfer. This is incorporated here into a modified version of the Marcus phenomenological theory relating reaction rate constants to equilibrium constants. It leads to Bronsted slope parameters that depend on the isotope transferred. The new theoretical formulation is tested on experimental data for hydride and deuteride transfer between nicotinamide adenine dinucleotide analogues and on computational data for hydrogen-atom and deuterium-atom transfer between pseudoatoms. The experimental kinetic isotope effects (KIE's) are shown to vary with reaction equilibrium constant (Kij) in a way that is quantitatively consistent with the theory. The critical configurations generated by the calculations vary from the saddle point and from each other in the way anticipated by theory. However, the calculated KIE values are a rather scattered function of because the tunneling corrections are large and somewhat system specific. Overall, we believe that this combination of experimental and calculated results provides considerable support for the idea that large-curvature tunneling needs to be considered in hydrogen-transfer reactions.
1. Introduction
In recent years there has been a rapid development of theory for hydrogen-transfer reaction~.I-~Theory now suggests that tunneling is significant in most hydrogen-transfer reactions (hydrogen atom, proton, and hydride ion) and that the heavy-atom framework in which this tunneling takes place is generally different from that which characterizes the saddle point of the potential energy surface connecting reactants and products.14 The geo(1) (a) Marcus, R. A,; Coltrin, M. E. J . Chem. Phys. 1977,67,2609. (b) Garrett, B. C.; Truhlar, D. G.; Grev, R. S.;Magnuson, A. W. J . Phys. Chem. 1980,84, 1730. ( c ) Truhlar, D. G.; Isaacson, A. D.; Skodje, R. T.; Garrett, B. C. J . Phys. Chem. 1982, 86, 2252. (d) Skodje, R. T.; Truhlar, D. G.; Garrett, B. C. J. Phys. Chem. 1981,85, 3019. J . Chem. Phys. 1982, 77, 5955. (2) (a) Ovchinnikova, M. Ya. Chem. Phys. 1979, 36, 85. (b) Babamov,
V. K.; Marcus, R. A. J . Chem. Phys. 1978, 74, 1790. (c) Babamov, V. K.; Lopez, V.;Marcus, R. A. J . Chem. Phys. 1983.78.5621. Chem. Phys. Lett. 1983, 202, 507. J . Chem. Phys. 1984,80, 1812. (d) Abusalbi, N.; Kouri, D.; Lopez, V.; Babamov, V. K.;Marcus, R. A. Chem. Phys. Lett. 1983, 203,458. (e) Nakamura, H.J . Phys. Chem. 1984,88, 4812. ( f ) Coveny, P. V.; Child,
M. S.;R h e l t , J. Chem. Phys. Lett. 1985, 220, 349. ( 9 ) Nakamura, H.; Ohsaki, A. J. Chem. Phys. 1985,83, 1599. (h) Babamov, V. K.; Lopez, V. J . Phys. Chem. 1986, 90, 215. (3) (a) Garrett, B. C.; Truhlar, D. G.; Wagner, A. F.; Dunning, T. H., Jr. J . Chem. Phys. 1983, 78, 4400. (b) Bondi, D. K.; Connor, J. N. L.; Garrett, B. C.; Truhlar, D. G. J . Chem. Phys. 1983, 78, 5981. (c) Garrett, B. C.; Truhlar, D. G. J . Chem. Phys. 1983, 79,4931. (d) Garrett, B. C.; Abusalbi, N.; Kouri, D. J.; Truhlar, D. G. J . Chem. Phys. 1985, 83, 2252. (4) See also: Marcus, R. A. J . Chem. Phys. 1966, 45, 4493; 1969, 49, 2617. Bowman, J. M.; Kuppermann, A.; Adams, J. T.; Truhlar, D. G. Chem. Phys. Lett. 1973, 20, 229. Kuppermann, A,; Adams, J. T.; Truhlar, D. G. Abstr. Pap. Int. Con$ Phys. Electron. At. Collisions 1973. 8, 149. Kuppermann, A. Theor. Chem. ( N . Y.)1981, 6A, 79.
metrical dislocation of the dominant tunneling path from the saddle point is particularly large for the case of a hydrogen, proton, or hydride ion being transferred between two much heavier atoms or groups; this may be considered to arise from the large curvature of the reaction path in mass-scaled coordinates in such ~ y s t e m s . ~ ? ~ In the present paper we attempt to adapt recent large-curvature-tunneling theory to exhibit some of its possible consequences for reactions of complex molecules in solution. In particular we focus on the consequences of the fact that, because tunneling is more facile for hydrogen than for deuterium, there are significant differences between the heavy-atom framework at the critical configuration for hydrogen transfer and that for deuterium transfer. Thus the critical configuration of a reaction-defined as the most probable structure for crossing the transition-state dividing surface that separates reactants and products-is different for hydrogen and deuterium transfer, and this isotopic shift can be treated analogously to a chemical substituent effect. For reactions of complex molecules in solution, reliable, realistic potential surfaces are not available, so comparison of calculated and observed rates of individual reactions does not provide a direct test of dynamical approximations. However, it is possible, especially for organic reactions in solution, to systematically vary the structure of the reactants and, thereby, of rate and equilibrium constants, by a series of very small steps, making the rate and equilibrium constants almost continuous functions of structural parameters. Thus it is possible to experimentally determine the derivatives of kinetic parameters with respect to the standard free energy of reaction, AGO, or, equivalently, with respect to the reaction equilibrium constant. It is attractive to test theoretical
0022-3654/86/2090-3766$01.50/00 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3767
Large-Curvature Tunneling in Hydride-Transfer Reactions models by comparing experimental and theoretical values of such derivatives, because the theoretical values of these derivatives may be less sensitive to the inevitable approximations and assumptions than the parameters themselve~.~In addition, the potential surfaces may be semiempirically adjusted to fit the value of the rate constant and whatever other parameter is being examined, making it more realistic. Because primary kinetic isotope effects (KIE's) are particularly sensitive functions of the tunneling contribution to reaction rates, we have chosen to apply dynamic tunneling theory by comparing the calculated and observed variation of such isotope effects under changes in AGO. Because of the anticipated differences between the critical configuration for hydrogen transfer and that for deuterium transfer, the KIE is predicted to change with the reaction equilibrium constant in ways that would not be expected in the absence of tunneling. A modified version of the Marcus theory of atom and group t r a n ~ f e r ~will - ~be ~ ~used ~'~ to systematize the data and facilitate the comparison. In Marcus theory the reactive event is divided into three stages: reactants to a precursor complex, precursor complex through an activated complex to a successor complex, and successor complex to products. The precursor and successor complexes, like the more familiar activated complex, may be conceptual constructs rather than real species. Further discussion of these complexes (and other aspects of Marcus theory) is provided elsewhere.8 In Marcus theory the free energy of activation for the second stage of reaction is modeled in terms of its overall free energy change and its so-called intrinsic barrier, and the latter is further estimated as the average of the intrinsic barriers for the symmetric analogue reactions. Thus Marcus theory is summarized by the following equations:
+ AjH AiH + Aj AG* = FV + (1 + (AGo'/X))'(X/4) AGO' = AGO - FV + WP Ai
-+
X = (hii
+ Xjj)/2
(1)
(2)
(3) (4)
Equation 1 represents a reaction series in which the hydrogen donor A, and/or the acceptor A, is varied. If an uncharged hydrogen atom is being transferred, all the participants may be neutral. If a proton or a hydride ion is being transferred, at least one product and one reactant are charged. (We limit ourselves to cases in which the charge types of the products are the same as the charge types of the reactants.) Since the ideas developed in this paper are applicable to hydrogen atom, proton, and hydride transfer, we omit charge designation in eq 1 and subsequently. AG* is The symbols of eq 2-4 have the following ~ignificance:~ the phenomenological free energy of activation defined by
AG* = -RT In [ k , , ( h / k T ) ]
(5)
where R and k are the molar and molecular Boltzmann constants, T i s the temperature, k,, is the reaction rate constant for reaction 1, and h is Planck's constant; W is the free energy of formation (from reactants) of the precursor complex; AGO' is the free energy change in proceeding from the precursor complex to the successor complex; h/4 is the intrinsic bamer; AGO is the free energy change in proceeding from reactants to products; WP is the free energy of formation of the successor complex from the products; and X,/4 and XU/4 are the intrinsic barriers for symmetrical cases, where A, is replaced by A, and vice versa. Note that eq 2 applies only to the case IAGO'l < A, which is satisfied for all cases considered in this article, and also note that AGO = -RT In K,,
(6)
(5) Hammett, L. P. Physical Organic Chemistry, 2nd ed.; McGraw-Hill: New York, 1970; Chapter 1 1 . (6) Marcus, R. A. J . Phys. Chem. 1968, 72, 891. (7) Hassid, A. I.; Kreevoy, M. M.; Liang, T.-M. Symp. Faraday SOC. 1975, I O , 69. (8) Kreevoy, M. M.; Truhlar, D. G. In Rates and Mechanisms of Reactions, 4th ed.; Bernasconi, C. F., Ed.; Wiley: New York, 1986; Chapter 1.
where Kij is the equilibrium constant for reaction 1. In the original Marcus theory6 W , WP, and X are constants, but we shall consider a modificationsJ9 in which X is a linear function of In Kij, or, equivalently, of AGO or AGO'. Marcus theory will be used to discuss two kinds of results. First we consider the variation in the KIE with changes in Kijfor hydride transfers among the nitrogen heterocycles 1-4. These compounds
R
R
;N7(
CH3L
4 ( A , L ; L = H or D )
are analogues of the enzymatic cofactor, nicotinamide adenine dinucletide (NAD'). Hydride-transfer reactions of this sort have been previously reported to give indications of t ~ n n e l i n g . ~ We J~ will also test the Marcus theory with semiclassical dynamic calculations of the KIE for a hydrogen-atom transfer between two pseudoatoms of mass 15, each roughly representing a methyl group, using a potential energy surface based on the extended London, Eyring, Polyani, Sato (LEPS) form," with an additional barrier term and with parameters adjusted to give AGO and AG* values similar to those determined experimentally for the hydride-transfer reaction, as well as a reasonable KIE, about 20, when AGO is zero. Rate constants for hydride transfer between analogues of NAD' have been shown to be quite well described by Marcus theory for a wide range of pyridinium, quinolinium, phenanthridinium, and acridinium At the present stage of data collection, 26 measured values of k,,, covering a range of 106.5,are available for comparison with Marcus theory. The average discrepancy between measured and calculated values is a factor of 1.5.13 Since most aspects of the theory are independent of charge type, we believe that the experimental and calculational results can be compared profitably. The use of a simple semiempirical potential surface and a three-body model system obviously represent a vast simplification of the experimental system, but we nevertheless believe that the model is adequate to represent some of the dominant factors controlling the experimental KIE's and their dependence on AGO. 2. Theory
It is customary and convenient to think about the dynamics of chemical reactions in terms of the motion of a mass point of unit or other fixed mass on a potential energy h y p e r s u r f a ~ e . ' ~ * ' ~ For the potential energy function used here, the minimum energy (9) OstoviE, D.; Roberts, R. M. G.; Kreevoy, M. M. J . Am. Chem. SOC. 1983, 105,1629.
(10) (a) Cook, P. F.; Blanchard, J. S.; Cleland, W. W. Biochemistry 1980, 19,4853, (b) Hermes, J. D.; Cleland, W. W. J . Am. Chem. SOC.1984, 106, 1263. (11) Kuntz, P. J.; Nemeth, E. M.; Polanyi, J. C.; Rosner, S. D.; Young, C. E. J . Chem. Phys. 1966, 44, 1168. (12) Roberts, R. M. G.; OstoviE, D.; Kreevoy, M. M. Faraday Discuss. Chem. SOC.1982, 74, 257. (1 3) OstoviE, D. Ph.D. Thesis, University of Minnesota, Minneapolis, 1985, pp 91-98. (14) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941; pp 100-107. (1 5) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics; Oxford University: New York, 1974; pp 98-100.
Kreevoy et al.
3768 The Journal of Physical Chemistry, Vol. 90,No. 16, 1986 24
CC d i s t a n c e ( A ) 32 36
28
40
,--. U v
x
‘
02 2.0
I
I
24
28
02 20
I
32
36
SO
-
mass of HC distance and y is the reactant H-to-C distance scaled to the same reduced mass. The top axis indicates the Ai-to-Cdistance and the long-short dashed curves are lines of constant A,-to-C distances of 2.7, 3.3, and 4.0 A. The solid curves are contours for 5 , 15, 25, 35, and 45 kcal/mol. The zero of energy is defined as the bottom of the reactant vibrational well. The saddle point is shown as a double dagger, and the minimum energy path is shown as a long-dashed curve. The vibrational cuts are shown as solid-linesegments; these are classical turning points in the vibrationally adiabatic potential curves for ni = 0 in the entrance channel and nf = 0, 1, and 2 in the product channel at a total energy of 20.92 kcal/mol. The small dots are the turning points in the stretching vibrational motion for the same total energy. The short-dashed curves are the large-curvature tunneling paths connecting initial state ni = 0 at its translational turning point to the final states nf = 0, 1, and 2 at theirs. In all cases (Figures 1-4) the total energy and nr values for which tunneling paths are_ shown are the total energy for which the integrand, p ( E ) exp(-E/kT), of the transmission coefficient peaks at 300 K and the three final states most populated by the tunneling process at this energy.
24
28
1.4
CC d i s t a n c e (A) 3.2 36
40
\
I
I
1
i
l
I
1.0
24
l4
28
06.
\,
1
28
I
I \
32
36
40
-
x (A) Figure 2. Same as Figure 1 except for the reaction A, HC HA, C with an exoergicity of 5 kcal/mol. The vibrational cuts are for a total energy of 19.39 kcal/mol.
+
36
40
-
(A)
C C d i s t a n c e (A) 32 36
40
r I
x
06
1 -
02 20
24
32 36 40 x (A) Figure 4. Same as Figure 1 except for the reaction A, + DC DA, + C with an exoergicity of 5 kcal/mol. The vibrational cuts are for n, = 0 and nf = 1, 2, and 3 at a total energy of 19.32 kcal/mol
28
-
COS’
h
I
32
Figure 3. Same as Figure 1 except for the reaction A, + DC DA, + C with zero exoergicity. The vibrational cuts are for a total energy of 20.90 kcal/mol.
5
24
I
I
28
(amu) as the mass of both A, and A,. These contour maps show that the skew angle, p, defined as the angle between the reactant and product valleys in a mass-scaled skewed-axis coordinate system, is fairly small, 20°, leading to large curvature of the MEP. In general the skew angle is given by”
h
20
\
x
x (-4)
Figure 1. Potential energy contours, tunneling paths, and vibrational cuts for the reaction Ai HC HA, + C with zero exoergicity. The mass-scaled coordinates are used as the axes: x is the Ai to center-of-
+
I
24
+
path (MEP) of reaction corresponds to the transferred H atom lying on a straight line from the acceptor, A,, to the donor, A,. In such a case of a collinear MEP, we also assume that the dominant tunneling paths are collinear. Two dimensions are sufficient to specify the internuclear distances on such paths, and if a third dimension is used for potential energy, the potential energy becomes an ordinary surface, which can be represented by a contour map and easily visualized. In a mass-scaled skewed-axis r e p r e s e n t a t i ~ n , ’ ~the ~ ’ ~ collinear motions of the three-atom system reduce to the motion of a mass point on this potential surface.14J5 The skewing eliminates cross terms in the kinetic energy expression, and the mass scaling makes it unnecessary to associate different masses with the moving point when the direction of its motion changes. Potential energy contour maps in such mass-scaled skewed-axis representations for the present system are shown in Figures 1 and 2 using 15 atomic mass units
P = mA,mAl/mA,HmA,H
(7)
where mx is the mass of X, so it is clear that the skew angle will be small whenever a light atom is transferred between two heavy atoms. As A, and HA, approach one another along the MEP, the H-A, stretching vibration is represented by a high-frequency oscillation of the mass point, transverse to this path. Each time the mass point approaches its outer turning point, there is a nonzero probability that it will tunnel through the barrier, cross the transition-state dividing ~ u r f a c eand , ~ ~appear ~ ~ ~in the product valley. This probability increases as the mass point approaches the saddle point (conventional transition state) of the surface. However, because a very large number of opportunities to react occur before the saddle is reached, the critical configuration, defined above, is reached before the saddle geometry. For a reaction that exhibits the tunneling behavior described above, the critical configuration is more extended than the conventional transition state. In such cases the critical configuration primarily determines the response of k,, to small changes in the reactivity of the reactants in the same way that transition-state structure determines that response’* when the transition state is the critical configuration. In the present case the critical configuration has a greater A,-A, distance than the conventional transition state, (16) Johnston, H . S. Gas Phase Reaction Rate Theory; Ronald: New York, 1966; p 121 ff. (17) Truhlar, D. G.; Garrett, B. C. Act. Chem. Res. 1980, 13, 440. (18) Reference 5 , pp 101-111.
The Journal of Physical Chemistry. Vol. 90, No. 16, 1986 3769
Large-Curvature Tunneling in Hydride-Transfer Reactions and we now explore the consequences of this difference for KIEs. If H is replaced with D, fl is larger, e.g., 28’ for transfer of D between two moieties of effective mass 15 amu, as shown in Figures 3 and 4. This arises not from a failure of the BornOppenheimer approximation, which is assumed to be completely valid here, but rather as a consequence of representing the dynamics of the system by the motion of a mass point in a massscaled, skewed-axis coordinate system. A consequence of the larger value of p is less extended-configuration tunneling, as expected when H is replaced with D. For this reason the critical configuration for the transfer of D is more similar to the transition state, which is the same on both surfaces, than that for the transfer of H is, and the A,-A, distance in the critical configuration for the transfer of D is smaller than that in the critical configuration for the transfer of H. In previous work19 it was shown that if k, and K,Jare varied by changing the structure of the acceptor, A,, the Bransted slope parameter a,defined by20 a = d(ln k,)/d(ln K,)
can be approximated in terms of two new parameters, by the relations a =x 0 . 5 ( ~- 1)
+
x = 0.5[1 - ( R T / X ) In K,]
(8)
x and T , (9)
(10)
- 40
0
-20
n!.
and 7
- 1 = d(ln k,,)/d(ln K , )
(11)
In this article we restrict our attention exclusively to structural variations in the acceptor; several signs must be changed to apply the theory when structural changes are made in the donors8 When 7 = 1, eq 8-10 are equivalent to eq 2, 3, 5, and 6 with constant X and constant, equal WT and WP, except that since eq 8 expresses the derivative of eq 2 it is missing a constant of integration. The introduction of T # 1 is equivalent to letting X vary with KIJ. The further a s s ~ m p t i o that n ~ ~T~is~constant makes X a linear function of In KIJ.The parameters x and 7 were rigi in ally'^-^' thought to depend on the symmetry and tightness, respectively, of the transition state. Following earlier precedents?l T was defined as the sum of the bond orders to the in-flight atom at the transition state. If the in-flight atom were completely detached from both AJ and A,, T would be zero. For a hypervalent in-flight atom, with full bonds to both AJ and A,, 7 would be two. If the bond order to the in-flight atom were conserved, 7 would be one. In view of the foregoing discussion, however, it is clear that x and T must refer to the critical configuration, rather than to the transition state, when these differ. Since it has been shown that the A,-A, distance of the critical configuration can be expected to be greater for H transfer than for D transfer and increasing this distance should decrease the total bond order to the in-flight atom, T should be smaller for H than for D transfer, and a,which is a measurable quantity, should also be smaller. Intuitively it seems likely that the difference between aHand aDwill be small. If that is the case, then, since a plot of Ink,, against InK,,, from which an experimental CY is obtained, shows scatter because of factors not considered in Marcus theory, as well as that due to experimental errors, the difference between the two a values may easily be less than their uncertainties. The effect of the difference of aH from aDis more significant for the KIE though. Therefore, we now derive an algorithm for the kinetic isotope effect of reaction 1, which we will call KIE,, as a function of KIJ. We consider only the case where the acceptor is varied to produce the changes in the measured quantities. Both AGH* and AGD* are calculated separately by using eq 2. We consider reactions for which A, is structurally and chemically similar to A,. As a consequence we assume that
w=WP=w
(12)
(19) Kreevoy, M. M.; Lee, S.4. H.J . Am. Chem. Soc. 1984,106,2550. (20) See, e&: Moore, J. W.; Pearson, R.G . Kinerics and Mechanism; 3rd ed.; Wiley: New York, 1981; p 353. (21) Albery, W. J.; Kreevoy, M. M. Adv. Phys. Org. Chem. 1978, 16,87.
20
35
Kij
Figure 5. Experimental and theoretical kinetic isotope effects (KIE,, equal to kij,H/kij,D) as functions of the natural logarithm of the equilibrium constant Kij,H. Experimental values are shown by circles. Theoretical values are shown as curves and are calculated from eq 2, 3, nnd 12-15 with W = 2.00 kcal mol-’,kH0/4 = 17.24 kcal mol-’, and h i 0 / 4 = 18.29 kcal mol-I. For curves 1 and 2, TH = T D and has the values 1 .OO (curve 1 ) and 0.77 (curve 2). For curves 3-5, TH remains 0.77, but T~ is 0.78 (curve 3), 0.79 (curve 4), or 0.80 (curve 5).
so that AGL” (L = H or D) equals AGLO, which is measurable. The variation in X with AGO’ has been shown to be given byI9 dX/d(ACo’) =
7
- 1
(13)
If we assume that X is a linear function of AGO’, we obtain X = Xo
+ [dX/d(AGo’)]AGo’
(14)
where Xo is the value of X for the member of the family of reactions which has AGO’ equal to zero. Since T has been shown to be reasonably constant,22eq 13 provides some support for the assumption that X is a linear function of AGO’. For hydride transfer, Xo and T were available from previous work.9J9,22 For deuteride transfer and for both sets of calculated atom-transfer rate constants, these parameters were obtained by fitting the data. These choices permit us to calculate AGL* for both H and D transfer, for both reaction families, as a function of Kij. The kinetic isotope effect KIEij is then given by kjj,H/kij,D = [exp(AGD*- AG,*)] / R T
(15)
which follows from eq 5 . The outcome, with Xo suitable for describing the experimental data, is shown in Figure 5 for various chcices of TH and T ~ .
3. Computations The potential energy surface for the three-body model reaction Ai
+ L-C
-
Ai-L
+C
(16)
where Ai is the acceptor group, L is H or D, and C denotes the methyl donor, is written V = V,LEPS+ Va
(17)
where VeLm is an extended LEPS function” and V, is an auxiliary barrier term. The extended LEPS term of the potential is completely specified by its diatomic Morse parameters-dissociation energy De, (22) Reference 13, p
90.
3770 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 equilibrium internuclear separation Re, and range parameter @ - a n d by its diatomic Sat0 parameters A. For the reactant, LC, we took De = 73 kcal/mol and Re = 1.1 A, and we set the range this yields parameter so that the force constant is 4.8 mdyn .k'; p = 2.175 A-1. For the product, AIL, De varies from 68 to 80 kcal/mol, Re is again set to 1.1 A, and p is again set to give a force constant of 4.8 mdyn A-l; this makes /3 depend on De. For example, when De is 68 kcal/mol, p is 2.254 k'. Both the L-C and A,-L parameters are typical of the carbon-hydrogen single bonds involved in the donors and acceptors studied experimentally in this For the nonbonded pair, A,-C, we set De = 80 kcal/mol and Re = 1.54 A, and we set 0to 2.078 A to again yield a force constant of 4.8 mdyn/A. The A,-C parameters are typical of a carbon-carbon bond. The range of dissociation energies given above yields a range of classical endoergicities, defined by A V = De(L-C) - D,(A,-L)
(18)
The other parameters of the family of potential energy surfaces were all taken to be independent of AV. The A,-L and L-C Sat0 parameters were set to zero, and the A,-C Sat0 parameter was varied until the classical barrier height, V ,equaled 18 kcal mol-' for the A V = -7 kcal mol-' case. This choice of barrier makes the model computational system similar to the experimental systems studied here, and it yields A(A,-C) = -0.189. The calculated kinetic isotope effects for this extended LEPS surface were deemed to be too large, and we attributed this to unrealistically narrow barriers. The LEPS model is known to give too narrow a barrier for the well-studied H + H2 reaction,23and in that case a better representation of the ab initio calculations is obtained by adding terms of the form24
Kreevoy et al. TABLE I: Barrier Height Quantities for the Potential Energy Surfaces Used for the Three-Bodv Modelo
-7 -5 -4 -1 0 2 3 5
18.5 19.1 19.5 20.6 21.1 22.1 22.6 23.8
AvaG= V v ~ + p AcstrC+ 2tb
(20)
where all quantities are evaluated on the minimum-energy path (MEP) and are functions of the distance along th,t path; VMEp is the Born-Oppenheimer potential, relative to reactants; AtstrG is the zero-point energy of a stretching motion perpendicular to the MEP (at the saddle point for the thermoneutral case this is the symmetric stretch) minus the zero-point stretching energy of reactants: and ebG is the zero-point energy of a single bending (23) Truhlar, D. G.; Wyatt, R. E. Adu. Chem. Phys. 1977, 36, 141. (24) Truhlar, D. G.; Horowitz, C. J. J . Chem. Phys. 1978,68, 2466; Errata 1979, 7 1 , 1514.
17.5 18.1 18.4 19.6 20.0 20.8 21.6 22.8
"All entries in kcal mol-'.
degree of freedom. For all cases considered here the value of AVaG at the saddle point is within 0.03 kcal mol-' of its maximum value, which is denoted AVZAGand which defines the location of the threshold variational transition state.'b.c AVaAGis the low-temperature limit of the activation energy of variational transition-state theory, and it is compared in Table I to the classical barrier height, which is denoted V and which is the maximum of VMEp.These quantities will be useful in interpreting the results (section 5). All rate constants for the atom-diatom model system were calculated by improved canonical variational-transition-state theory (ICVT)Ib with least-action ground-state (LAG)3c,dtransmission coefficients to account for tunneling. The final bimolecular rate constant is given by the expression
where the "quasiclassical rate constant" kTCVT corresponds to classical reaction-coordinate motion with other degrees of freedom quantized and the transmission coefficient K~~~ accounts for quantum effects on the reaction coordinate, primarily tunneling. The transmission coefficient involves a thermal average K~~~
where R,,R2, and R3 are the internuclear distances and n 1 2 (to avoid the necessity of nonanalytic absolute values, n should be even). This term is very convenient for widening barriers since it vanishes when any two internuclear distances are equal or when any one is large; thus it can be used effectively to put "shoulder pads" on a symmetric or near symmetric barrier, such as the ones involved in the present study. We added a single term of the form (19) to the extended LEPS surface described above to widen the barrier and, presumably, make it more realistic. We chose n = 2, the simplest allowed value, and we set a = 0.004 ao-), which is one significant figure is the same as was used for H + H2. Finally a was increased until the calculated KIE for the A V = 0 case at 300 K was only about 20. This yields a = 0.026 Eh sod. (Note: the parameters are given in atomic units defined by 1E , = 627.5 kcal mol-', la, = 0.5292 A.) For the purpose of characterizing the shapes of the potential energy barriers, we note that for transfer of H the imaginary frequency of the saddle point increases from 15 14i cm-I at A V = -7 kcal mol-] to 17581' cm-I at A V = 0, and then it decreases to 1677icm-I at A V = +5 kcal mol-'. The imaginary frequencies for D transfer are smaller, in the range 1157i-12641' cm-'. The surfaces may also be characterized in terms of their vibrationally adiabatic ground-state potential curves, defined bylb.c
16.9 17.5 17.9 19.0 19.5 20.5 21.0 22.2
= lR:dE p ( E ) exp(-E/iT)
where E is the total energy, cRG is the zero-point energy of reactants, and p ( E ) is the ground-state tunneling probability summed over final vibrational states. The LAG method includes large-curvature tunneling as a special case,3c,25 but should be more accurate than our original large-curvature a p p r o x i m a t i ~ n in ~~,~ the general case. Rotational and vibrational partition functions were approximated by the independent-mode approximation, with rigid rotations and anharmonic vibrations. Anharmonicity of stretching vibrations was treated by the Wentzel-Brillouin-Kramers (WKB) method,26and quartic anharmonicity was included for the bend by a variation-perturbation method, with the quartic bend force constant obtained by a Taylor series expansion in deviations from the linear geometry with the same nearest-neighbor distances.Ib The LAG transmission coefficients, which are the same for the forward and reverse reactions, are always calculated in the exothermic direction of reaction with ground-state reactants and summed over final vibrational ~ t a t e s . All ~ ~ .final ~ vibrational states energetically allowed at the transition-state threshold energy are included; this implies 3-4 states for L = H and 4-5 states for L = D. All calculations on the three-body model system are for a temperature of 300 K. 4. Experimental Section
Compounds 1-4 were prepared by methods that have been previously d e s ~ r i b e d . ~The - ~ ~dideuterated ~~~ variant of 4 was prepared by reduction of the corresponding amide with LiA1D4 (25) Truhlar, D. G.; Isaacson, A. D.; Garrett, B. C. In Theory of Chemical Reaction Dynamics; Baer, M . , Ed.; CRC: Boca Raton, FL, 1985; p 65. (26) Garrett, B. C.; Truhlar, D. G. J . Chem. Phys. 1984, 81, 309. (27) Roberts, R. M. G.; OstoviE, D.; Kreevoy, M. M. J . Org. Chem. 1983, 48, 2053. (28) OstoviE, D.; Lee, 1 . 6 . H.; Roberts. R. M. G.; Kreevoy, M. M. J . Org. Chem. 1985, 50, 4206.
Large-Curvature Tunneling in Hydride-Transfer Reactions (Aldrich Chemical Co., 98% D). The mass spectrum of 4 was consistent with dideuteration, but no quantitative limit could be placed on its content of monodeuterated variant because of the very strong tendency of the parent ion to fragment by loss of H or D, even a t minimal ionizing voltage. Visual inspection of the N M R spectrum of 4 gave no clear evidence of the presence of undeuterated material. However, electronic integration of the region of the spectrum corresponding to the undeuterated methylene group gave 4.5% of the intensity of the benzyl methylene group. This gives a 9% upper limit for the contamination of dideuterio variant with monodeuterio variant. Rates were measured in a solvent containing four parts of 2-propanol to one part of water, by volume, acidified to pH 4-4.5 with HC104 to avoid the hydroxylation of the quinolinium ions.28 Pseudo-first-order rate constants, kij('),were measured spectrophotometrically by monitoring the appearance of the phenanthridinium ions a t -375 nm. The quinolinium ions are always in excess by a t least a factor of 15, and equal quinolinium concentrations were used for ki,,Hand kij,D.In all cases the absorbance A, was monitored until it was stable; usually A , - A. was 0.2-0.3. The rate constant is given byz9
The data for a reaction extent of about 95% was then processed with a computer program that adjusted both A , and kl so as to minimize C [ A ,- A,(calcd)12, where A,(calcd) is a value of A, calculated from eq 23, using trial values of kij(')and A,. The difference between the observed values of A , and those selected by the program were never more than 0.002. The temperature of the reaction mixtures was controlled by circulating controlled-temperature water through the walls of the cell compartment. It was measured in a spectrophotometer cell containing only solvent, in the cell compartment with the reaction mixtures. It was, typically, within 0.1 of 25.0 O C and was never deviant by more than 0.2 O C . Second-order rate constants, kij,were obtained by dividing kij(') values by the quinolinium ion concentrations. Our spectrophotometer cell compartment had room for four cells: a reference cell containing the mixed solvent; a reaction mixture including a phenanthridan; another reaction mixture, similar except that the phenanthridan was replaced with the corresponding dideuteriophenanthridan; and the cell for measuring temperature. In this way, the influence of temperature fluctuations on the measured KIEijvalues, k,,/k,,, was largely eliminated by taking ratios of rate constants measured at the same time and ultimately averaging the KIEijvalues instead of averaging the kijvalues first and then getting the KIEijvalues by taking ratios of average kij values. Each KIEij was measured at least four times, more typically six times. The average deviations from the mean values were 2-3%, and the standard errors of the mean values were 1%. If 4 was contaminated with 9% of the monodeuterio compounds, the initial values of kij,D(')would have been too large by 18%. However, since the monodeuterated contaminant would have been consumed over twice as fast as the dideuteriophenanthridan, ki,,D(') would have decreased steadily during the course of an experiment, ultimately approaching the true value. Small deviations from the expectations of eq 23 would have resulted, but is is not clear that these could have been detected. In this worst case, the resulting values of kij,D(l)and kij,Dwould probably be 10% too large and the KIEij values about 10% too small. However, the trends in KIEij, reported in the Results section, would have been unaltered since only two samples of the dideuteriophenanthridan were used and they were prepared in identical ways.
-
-
-
5. Results
The experimental results are summarized in Table 11. The rate constants for deuteride transfer are not shown but may be obtained by dividing k , , by KIE,. The rate constants and kinetic isotope effects typically have standard errors of 1%; no standard error is larger than 2%. The equilibrium constants refer to re(29) Frost, A. A.; Pearson, R.G. Kinetics and Mechanism, 2nd ed.;Wiley: New York, 1961; p 29. Reference 20, p 19.
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3771 TABLE 11: Experimental Kinetic Isotope Effects on Hydride Transfer Ai In Kij,*" k.. u,H* M-1 s-1 KIEjj la 6.86 0.1623 5.48 Ib
IC Id le If
8.37 10.01 17.32 17.66 18.98
0.424 0.735 6.95 7.42 9.31
5.40 5.22 4.91 4.84 4.60
uCalculated from equilibrium constants given in ref 19 and 28. actions of the undeuterated variant of 4. Although the equilibrium isotope effects have not been measured, it is quite likely that they deviate negligibly from unity. Figure 5 compares these results to several of the curves of KIEi, as a function of Kij for various choices of the parameters in eq 2, 3, and 12-14. As shown, the experimental results fall almost perfectly on one of these curves, which requires slightly different T values for transfer of H and D. To generate the curves shown in Figure 5, ?Vand WP were both taken as 2 kcal mol-'. As noted above, the reactions are structurally very similar in the forward and reverse directions, SO it is reasonable that WP, which is the analog of W for the reverse reaction, should be the same as W . Beyond that, the exact values chosen have very little effect on the KIE, values as long as they are small compared to X/4, which they appear to be.I3 To get a value for AHo, Xii was first obtained for a hypothetical member of series 1, which would have the same reduction potential as 3. This was done by using eq 11 with previously available values of kii for members of this series'2*22and using a value of 0.77 for rH. From this kii a value for AGii* was calculated by an equation equivalent to eq 5 , and then a Xii was obtained by subtracting 2 kcal mol-' from AG*. This Xii now refers to an acceptor that will have an equilibrium constant of 1.0 when it reacts with 4. Then the known'* X value for the 3-4 couple was combined with Xii in eq 4 to give a Xij value, which is AHo, that is, the X value for the reaction of the type of eq 1 with Kij = 1.0. The value of XH0/4 obtained is 17.24 kcal mol-'. Except for one curve illustrating the effect of taking T , and rDas 1.00, the previously determined12I9value of 0.77 was used for rH. ADo and rDwere adjusted to fit the present data. Modest variations of AHo and ADo have little effect on the shape of the curves, but ADo - AHo determines the position of the curve on the vertical coordinate. The curvature and location of the maximum are largely determined by rD- rH. Table I11 shows similar quantities, Kij,H,kij,,, and KIEij, as in Table 11, but for the three-body model problem. The table also shows the calculated transmission coefficients for both H and D. The transmission coefficients account for quantum effects on reaction-coordinate motion and are greater than unity because of tunneling. On the series of potential energy surfaces used here, tunneling is important for all values of AV, even for deuterium transfer. The inclusion of transmission coefficients in Table I11 allows us to calculate what the kinetic isotope effects would be in the absence of tunneling; we obtain values in the range 4.3-4.6 in all cases. The most striking result of the calculations on the three-body model is that the critical configurations are indeed significantly more extended than the saddle point. For the model potential energy surfaces studied here, the variational transition state is at the saddle point for both isotopes and all AVconsidered, but the critical configuration is more extended for H than for D because of tunneling. The observation that the critical configurations are more extended than the saddle point is illustrated by any of Figures 1-4, which show the least extended tunneling paths (those terminating at translational turning points) for the energies that dominate the thermal average. The isotope dependence of the critical configuration is seen clearly by comparing Figure 1 to Figure 3. The isotope effect on the donor-acceptor distance at the critical configuration for nonthermoneutral reactions is not clear in Figures 2 and 4, but would probably emerge from a complete thermal average over the wide distribution of tunneling paths corresponding to various termini of the tunneling paths along
3772 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
Kreevoy et al.
TABLE 111: Results for the Three-Bodv Model of H and D Transfer -7 -5 -4 -1 0 2 3 5
1.2 4.4 8.1 5.4 1 .o 3.5 6.5 2.3
x 105 x 103 x 102
23.8 24.5 29.9 81.3 90.9 66.1 45.8 28.0
x 10-2 x 10-3 x 10-4
14.8 15.3 16.1 21.4 19.2 20.7 20.0 16.8
1.2 4.3 3.0 1.1 5.9 8.1 2.2 1.9
x 10-1 x 10-2
7.4 7.0 8.2 16.4 20.5 13.6 9.9 7.3
X
x 10-2 x lo-' x 10-4
x 10-4 x 10-5
TABLE IV: Best-Fit Parameters of the Marcus Equation of the Three-Body Model Problem
a
rate const
w
An"
7
k(0' kH kD
16.6 18.5 18.5
18 10 16
0.93 0.87 0.92
kcal/mol.
the MEP and to a Boltzmann distribution of total energies and a sum over final vibrational states. The curvature of a Bransted plot based on the calculated rate constants is much larger than anticipated by eq 2, 12, and 14 if W is set equal to zero. Although W = 0 might seem reasonable as there are no wells, barriers, or special orientation or desolvation requirements in the entry or exit channel, in order to fit the results to the modified Marcus theory we have adjusted W. Table IV lists the values of W, A', and T which give best fits of the results to the theory for zeroeth-order rate constant (the same for hydrogen and deuterium transfer) and for the final (ICVT/LAG) hydrogen- and deuterium-transfer rate constants. The zeroethorder rate constant is simply
and is considered only to elucidate the significance of the large Wvalues. The fit obtained for kijc0)is shown in Figure 6, and Table IV shows that even this simple rate constant requires a large Win the modified Marcus equations. Thus the large W is not a manifestation of tunneling but rather follows from the way the barrier height depends on A V for our series of surfaces. With W = 18.5 kcal mol-', an excellent correspondence with the modified Marcus theory is obtained for deuterium transfer, as shown in Figure 7, but the correspondence is somewhat worse for hydrogen transfer. For both isotopes the transmission coefficients are largest in the vicinity of Kij = 1.0, and the fit, particularly for hydrogen transfer, is correspondingly poorest there. However, the best-fit T values show the anticipated trend: do) > T~ > T ~ From . the values of the parameters given in Table 111 for kij,Hand kij,D,KIEij can be calculated as a function of Kij, as described in the theoretical section. Figure 8 compares the calculated (Table 11) and theoretical values. The correspondence is not good because the large transmission coefficient incorporated in kij,Hdoes not vary with Kij in the way predicted by either the original or the modified Marcus theory. However, as with the experimental and modified Marcus equation KIE values, the ICVT/LAG calculations show a strong tendency for the KIEij to fall with increasing Kij for Kij > 1.0. 6. Discussion
The series of potential energy surfaces we have used is not unreasonable. It gives realistic values of reactant and product vibrational frequencies (3000 cm-I) and dissociation energies (68-80 kcal mol-'; considerably lower than a typical C-H bond energy30because we were attempting to mimic a particularly weak C-H bond), and the resulting constants are of similar magnitude to those observed for hydride-transfer reactions. Nevertheless, we cannot claim that the surfaces are realistic. A wide variety of other surfaces would meet the criteria we haTre mentioned. The (30) Benson, S. W. Foundations of Chemical Kinetics; McGraw-Hill: New York, 1960; p 670.
-12
1 i n Kij
Figure 6. Brernsted plots for the zeroth-order rate constant of the three-body model problem as a function of equilibrium constant. Also shown is the Marcus equation fit with the parameters of Table IV.
1 4 r ' I -10 -6
'
I
I
-2
2
in
'
'
6
'
10
'
I
14
I
Kij
Figure 7. Same as Figure 6 except for ki,,H and k i , , ~calculated by the
ICVT/LAG method.
fact that the calculated values of k , require a very high value of Win order to be fitted to eq 2, even when all the dynamic factors are omitted (Figure 6 ) , is a warning that the potential surfaces used for the calculations differ in important ways from the real surfaces since the measured rate constants require only modest Wvalues. The potential surfaces for the actual organic systems studied experimentally here have higher dimensionality than those for the three-body model problem, both to represent the many additional vibration modes of the donors and acceptors and also to describe the many solvent modes that must participate in the reaction coordinate. Some of these modes have weak restoring forces and many closely spaced energy levels, so that the small number of state-to-state tunneling possibilities available to the model systems would be replaced with a much larger variety of possibilities. This would tend to smooth out the irregularities in the Ink, vs. lnKij plots (these are particularly visible in the kij,H plot of Figure 7 ) . Participation of other modes in the reaction
Large-Curvature Tunneling in Hydride-Transfer Reactions
0
0
in Kij Figure 8. Kinetic isotope effects corresponding to the data and curves in Figure 7.
coordinate might also raise the effective mass for motion along the reaction coordinate and thus reduce the difference in effective mass between the hydrogen-transfer and deuterium-transfer variants. This change would tend to reduce the kinetic isotope effect. Thus the calculations are not reliable, a priori predictions of the rate of any real chemical reactions nor are they meant to be, especially since the development of realistic potential energy surfaces for reactions of many-atom substances remains a work for the future. Nevertheless, the calculations do give an indication of how tunneling occurs and what its observable consequences are when it does occur in real systems. The indications are that corner cutting is quite significant and that it is more important for H than for D. As expected, more corner cutting is associated with lower T values, as shown in Table IV. The calculated kinetic isotope effect shows the expected, steep fall as Kij increases from 1.0 (Figure 8), but, in fact, it is a much steeper function than anticipated by eq 2-5 and 8-15 because of the large K~~~ values for H transfer when Kij is near 1.0. However, this discrepancy may be, at least partially, an artifact of the simplicity of the system for which calculations were made, as described above. In a qualitative way there is a good match between the behavior of the experimental system (Figure 5) and the model system. We believe that this correspondence strongly suggests that largecurvature tunneling with corner cutting is an important feature of hydride transfer between carbon atoms, and probably of many other hydrogen-transfer reactions as well. Examples from the l i t e r a t ~ r e ~where ' , ~ ~such ~ effects may be considered are discussed next. Stewart and Toone3I have investigated hydride transfer to 'R from formate ion or some other derivative triarylcarbocations,, of formic acid formed in trifluoroacetic acid, and they found a relatively sharp kinetic isotope effect maximum (as a function of the pKR+values of the carbocations3*). The mechanism of these reactions and the real hydride donor are somewhat uncertain. It has been suggested43that the carbocations are first esterified by the formate ion, and these formate esters undergo an internal rearrangement to form the products. Hydride transfer from formic acid may be still another possibility. However, in themselves,
+
(31) Stewart, R.; Toone, T. W. J. Chem. Soc., Perkin Trans. 2 1978, 1243. (32) KR+is the equilibrium constant for the reaction, R+ + H20* ROH H+. (33) Olah, G. A,; Svoboda, J. J. J . Am. Chem. SOC.1973, 95, 794.
7'he Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3773
neither of these explains the sharp maximum in the kinetic isotope effect, which may be due to large-curvature tunneling. Stewart and Toone fitted the logarithms of their rate constants to a quadratic function of the logarithm of an equilibrium constant assumed to correlate with Kij. Quadratic free energy plots, as described elsewhere,s are equivalent to eq 2 with 7 = 1.0; the equivalence yields A E 6 kcal mol-]. This is surprisingly small in consideration of the smallness of the measured rate constants and the large magnitude of the apparent heat of activation, which is reported3' to be 27 kcal mol-' for a typical member of the reaction series. Their plot of experimental KIE,, values vs. pKRt differs from their best quadratic fit in having a narrower peak in qualitatively the same way that our curves 3-5 in Figure 5 differ from the quadratic fit, curve 1, in that plot. It seems likely that the present theory, with T~ # T D , is capable of producing a KIEij vs. Ki, plot similar in appearance to the experimental results of Stewart and Toone while maintaining a larger value of A. The use of a value of T other than unity would also permit the plot of log kH vs. -pKR+ to be fitted with a larger value of X,8 although the best-fit value of X would probably remain small, compared to the apparent activation energy. This interpretation supports and elaborates the suggestion of Stewart and Toone3' that tunneling might be involved in their reactions. Another set of experimental results that may be explained in terms of the fact that large-curvature tunneling leads to a more expanded critical configuration for hydrogen transfer than for deuterium transfer involves kinetic isotope effects in reactions where the approach to the transition state is sterically hindered. When the KIE is measured in a series of related reactions, it is a common o b s e r v a t i ~ n that ~ ~ - its ~ ~value is substantially higher for examples showing such steric hindrance than for the others. This is understandable if the reaction partners must approach each other more closely in order to transfer deuterium. Our theory may also help to explain the variation with pressure of large primary hydrogen K I E s that are thought to be partly due to t ~ n n e l i n g . ~ ~An - ~ example ' is the abstraction of a proton from 2-nitropropane by 2,4,6-trimethylpyridine in a tert-butyl alcohol-water mixture. The KIE for this reaction has been reported to fall from 1644to less than 8 as the pressure is increased from 1 bar to 1.3 kbar.39 When this result is treated by conventional TST. A P for hydrogen transfer is smaller than AV for deuterium transfer by 9 cm3 This would appear to represent a substantial breakdown of the Born-Oppenheimer approximation and conventional transition state theory, according to which the structures of the two conventional transition states are the same. Our present theory resolves the difficulty by providing critical configurations that are different from the transition state in such a way that the hydrogen one is more expanded than that for deuterium transfer. The real situation involves additional complications in that the KIE's themselves and their pressure derivatives are sensitive to solvent and the details of s t r ~ c t u r e . ~ ' Nevertheless the relative volumes of the critical complexes are in accord with the theoretical model. The theory also may be used to suggest other small differences between critical complexes for (34) Wiberg, K. B. Physical Organic Chemistry; Wiley: New York,1964; p 143. (35) Manka, M. J.; Stein, S. E. J . Phys. Chem. 1984, 88, 5914, and references therein. (36) Funderburk, L.; Lewis, E. S. J . Am. Chem. SOC.86, 2531 (1964). (37) Hanna, S. B.; Jermini, C.; Zollinger, H. Tetrahedron Lerr. 1969, 4415. (38) Colle, T. H.; Lewis, E. S. J . Am. Chem. SOC.1979, 101, 1810. (39) Isaacs, N.S.; Javaid, K.; Rannala, E. J . Chem. SOC.Perkin Trans. 2, 1978, 709. (40) Isaacs, N.S.; Javaid, K. J . Chem. SOC.,Perkin Trans. 2, 1979, 1583. (41) Sugimoto, N.; Sasaki, M. J . Chem. SOC.,Faraday Trans. 1 1985.81, 2959. (42) KR+is the equilibrium constant for the reaction, R+ + H 2 0 ROH H+. (43) Olah, G. A,; Svoboda, J. J. J . Am. Chem. SOC.1973, 95, 794. (44) Values of 24 and 20 have also been reported for this KIE: Lewis, E. S.;Funderburk, L. H. J . Am. Chem. SOC.1967,89, 2322. Bell, R. P.; G d a l l , D. M. Proc. R. SOC.London, A 1966, 294, 273. However, to see the effect of pressure it is better to compare a single, internally consistent, set of measurements.
+
J . Phys. Chem. 1986, 90, 3774-3783
3774
hydrogen transfer and those for deuterium transfer, without invoking a breakdown of the Born-Oppenheimer approximation.
Acknowledgment. The work of M.M.K. and D.O. was supported in part by the National Science Foundation through Grant No. CHE85-15014, that of D.G.T. was supported in part by the
U S . Department of Energy through Contract No. DE-AC0279ER10425, and that of B.C.G. was supported by the U.S. Army Research Office under Contract No. DAAG-29-84-C-0011. Registry No. l a , 8481 1-85-8; lb, 47072-02-6; IC, 102808-44-6; Id, 89321-40-4; le, 85289-84-5; If, 89321-43-7; D2, 7782-39-0.
Isotope Effects in Double Proton Transfer Reactions W. John Albery Department of Chemistry, Imperial College, London SW7 ZAY, England (Received: February 5, 1986)
Kinetic isotope effects are calculated for a reaction involving the simultaneous transfer of two protons. A linear transition state of two protons between three heavy entities of equal mass is assumed. Expressions are obtained for the four frequencies associated with motion along the line of centers. The variation in the size of these frequencies and their associated amplitudes as a function of the coupling of the protonic motion is explored. As the coupling is increased, one finds possible transition states where the reaction coordinate involves heavy-atom motion rather than a protonic vibration; such a transition state could exhibit a kinetic isotope effect of 2 to 3. Kinetic isotope effects larger than 20 are unlikely. The breakdown of the rule of the geometric mean for successivedeuterium substitutions is explored; the largest breakdown is found for a concerted symmetrical transition state and may be as large as 15%. The breakdown of the Swain-Schaad relation is smaller; for a symmetrical transition state it may be 5%.
Introduction In reactions that involve the transfer of two protons, an interesting question is whether there are two protons in flight in a single transition state or there are two successive transition states in which each proton is transferred one at a time. Examples of this type of system are numerous and range from classical reactions such as the enolization of acetone' and proton motion in the aqueous solvent,*through transfers in cyclical systems investigated by L i m b a ~ h ,to ~ .transfers ~ in biochemical systems5 and in particular in enzyme-catalyzed reactions. An example of such an enzyme-catalyzed reaction is the racemization of proline (I) P
C
O
2
H
so that the isotope effect observed for substitution on the a site will not be the same in the two cases. We have described elsewhere the detailed methodology of this approach." This method was also devised independently by Cleland and his group and applied by them to a number of enzyme-catalyzed reactions.'*I2 However, the success of the method for double-proton transfers depends on whether the rule of the geometric mean holds for the successive substitutions in the single concerted transition state. In this paper we explore this question. Another approach to the same problem, proposed by Northrop,I3 is to use the breakdown of the Swain-Schaad re1ati0n.l~ The Swain-Schaad relation relates the tritium and deuterium isotope effects:
I
H
I
catalyzed by the enzyme proline racemase. We have recently established the complete mechanism and free energy profile for this reaction.6 In order to answer the question posed above, we compared the kinetic isotope effect by substituting D for H on the first (a)site with H on the second (8) site with the kinetic isotope effect from the same substitution of D for H on the a site but this time with D on the (3 site;'g we compared kD,H/kH,H with kD,D/kGH,D. For a single concerted transition state, where the rule of the geometric mean9 is obeyed, we would expect that these two isotope effects would be the same. On the other hand, for two consecutive transition states the substitution of D for H on the /3 site changes the relative free energies of the two transition states ~~~
Northrop argued that for a single transition state the relationship should be obeyed, while for two concerted transition states there would be a breakdown in the relation because again the contributions of the two partially rate limiting transition states would be different for the deuterium and for the tritium substitution. We have argued that the deviations are small and therefore the method requires data of the highest p r e ~ i s i o n . ' ~In this paper we explore the breakdown of the Swain-Schaad relation which arises in a single transition state from the coupled motion of the two protons. The third question that we address is the size of the isotope effect to be expected in a single concerted transition state. This has been a point of controversy for some time.I6J7 One might argue that because two protons have lost their zero point energy and are flying in the transition state one would expect a larger
_____
(1) Albery, W. J. Faraday Discuss. 1982, 74, 245. (2) Albery, W. J. Prog. React. Kinet. 1967, 4, 355.
(3) Limbach, H. H.; Hennig, J.; Gerritzen, D.; Rumpel, H. Faraday Discuss. 1982, 74, 229. (4) Geritzen, D.; Limbach, H. H. J . Am. Chem. SOC.1984, 106, 869. ( 5 ) Gandour, R. D.; Schowen, R. L. Transition States of Biochemical Processes; Plenum: New York, 1978. (6) Fisher, L. M.; Albery, W. J.; Knowles, J. R . Biochemistry 1986, 25, 2529. (7) Belasco, J. G.; Albery, W. J.; Knowles, J. R. J. Am. Chem. Soc. 1983, 105, 2475. ( 8 ) Belasco, J. G.; Albery, W. J.; Knowles, J. R. Biochemistry 1986, 25, 2552. (9) Bigeleisen, J. J . Chem. Phys. 1955, 23, 2264.
0022-3654/86/2090-3774$01.50/0
(IO) Hermes, J. D.; Roeske, C. A,; O'Leary, M. H.; Cleland, W. W. Biochemistry 1982, 21, 5106. (11) Hermes, J. D.; Morrical, S.W.; O'Leary, M. H.; Cleland, W. W. Biochemistry 1984, 23, 5479. (12) Hermes, J. D.; Tipton, P. A,; Fisher, M.A.; OLeary, M. H.; Morrison, J. F.; Cleland, W. W. Biochemistry 1984, 23, 6363. (13) Northrop, D. B. Biochemistry 1975, 14, 2644. (14) Swain, C. G.; Stivers, E. C.; Reuwer, J. F.; Schaad, L. J. J . Am. Chem. SOC.1958,80, 5885. (15) Albery, W. J.; Knowles, J. R. J. Am. Chem. SOC.1977, 99, 637. (16) Bell, R. P.; Critchlow, J. E. Proc. R . SOC.London, A: 1971,325,35. (17) Bell, R. P. The Tunnel Effect in Chemistry; Chapman and Hall: London, 1978; p 90.
0 1986 American Chemical Society
