J . Phys. Chem. 1991, 95, 10425-10431
10425
Feynman Path Integral Approach for Studying Intramolecular Effects in Proton-Transfer Reactions Daohui Li and Gregory A. Voth* Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania I9104-6323 (Received: April 8, 1991)
A Feynman path integral methodology is developed to study proton-transfer reactions in polar solvents. The present approach is specifically designed to efficiently probe the effects of intramolecular vibrations on the transfer rate as a general function of temperature and intramolecular couplings. A representativestudy of some nonlinear couplings which may affect the tunneling of the proton will be presented. A central feature of the method is that the actual intramolecular proton-transfer coordinate is taken as the reaction coordinate. Even so, the activated dynamics of the solvent polarization is shown to be accounted
for.
I. Introduction Proton-transfer reactions in condensed phases are of substantial importance in chemistry. These reactions have attracted a considerable degree of experimentall-' and t h e o r e t i ~ a l ~attention -'~ in recent years. To the experimentalist and theorist alike, proton transfers are particularly interesting because they can have a high degree of quantum mechanical character. Since these reactions also involve a substantial redistribution of solute electronic charge density, it has been noted by several that the dominant (1) Bell, R. P. The Proton in Chemistry, 2nd ed.; Chapman and Hall: London, 1973; The Tunnel Effect in Chemistry; Chapman and Hall: London, 1980. (2) Klopffer, W. Ado. Photochem. 1977, 10, 3 11. (3) Huppert, D.; Gutman, M.;Kaufman, M. J. A d a Chem. Phys. 1981, 47, 643; Hibbert, F. Ado. Phys. Org. Chem. 1986, 22, 113. (4) Kasha, M. J. Chem. Soc., Faraday Trans. 2 1986,82, 2379. (5) Barbara, P.F.; Jarzeba, W. Acc. Chem. Res. 1988, 21, 195. (6) See, for example: Strandjord, A. J. G.; Barbara, P. F. Chem. Phys. Lett. 1983,98,21; J. Phys. Chem. 1985,89,2355; 1985,89,2362. Barbara, P. F.; Walsh, P. K.; Brus, L. E. J . Phys. Chem. 1989, 93,29. Fuke, K.; Yabe, T.; Kohida, T.; Kaya, K. J. Phys. Chem. 1986, 90, 2309. Limbach, H. H.; Henning, J.; Gerritzen, D.; Rumpel, H. Faraday Discuss. Chem. SOC.1982, 74, 822. Brunton, G.; Gray, J. A,; Griller, D.; Barclay, L. R. C.; Ingold, K. U. J. Am. Chem. SOC.1978, 100,4197. Loth, K.; Graf, F.; Giinthard, Hs. H. Chem. Phys. 1976, 13, 95. McMorrow, D.; Kasha, M. J. Phys. Chem. 1984, 88, 2235; J. Am. Chem. SOC.1983, 105, 5133. Woolfe, G. J.; Thistlewaithe, P. J. J . Am. Chem. SOC.1981, 103, 691. Brucker, G. A.; Kelley, D. F. J. Phys. Chem. 1987, 91, 2856; 1988, 92, 3805. (7) The entire issue of Chem. Phys. 1989,136, No. 2 is devoted to proton transfer and contains many additional references. (8) Dogonadze, R. R.; Kuznetzov, A. M.; Levich, V. G. Electrochim. Acta 1968, 13, 1025. German, E. D.; Kuznetzov, A. M.; Dogonadze, R. R. J. Chem. SOC.,Faraday Trans. 2 1980, 76, 1128. (9) Briinische-Olsen, N.; Ulstrup, J. J. Chem. SOC.,Faraday Trans. 1 1979, 75, 205. (10) Benderskii, V. A.; Goldanskii, V. I.; Ovchinnikov, A. A. Chem. Phys. Lett. 1980,73,492. Goldanskii, V. I.; FIeurev, V. N.; Trakhtenberg, L. I. SOD. Sci. Rev.B., Chem. 1987, 9, 59. (11) Trakhtenberg, L. I.; Klochikin, V. L.; Pshezhetsky, S. Ya. Chem. Phys. 1982.69, 121. (12) Siebrand, W.; Wildman, T. A.; Zgierski, M. Z . Chem. Phys. Lett. 1983, 98, 108; J. Am. Chem. SOC.1984, 106,4083; 1984, 106, 4089. (13) Meyer, R.; Ernst, R. R. J. Chem. Phys. 1987,86, 784. Okuyama, S.; Oxtoby, D. W. Ibid. 1988, 88, 2405. (14) (a) Borgis, D.; Hynes, J. T. In The Enzyme Catalysis Process; Cooper, A., Ed.; Plenum: New York, 1989. (b) Borgis, D.; Lee, s.;Hynes, J. T. Chem. Phys. Lett. 1989, 162, 12. Borgis, D.; Hynes, J. T. J. Chem. Phys. 1991, 94, 3619. (15) (a) Cukier, R. I.; Morillo, M. J. Chem. Phys. 1989, 91, 857. (b) Morillo, M.; Cukier, R. 1. Ibid. 1990, 92, 4833. (16) (a) Warshel, A. J. Phys. Chem. 1982,86,2218; (b) Proc. Natl. Acad. Sci. U.S.A. 1984.81.444. (c) Warshel, A.; Chu, 2. T. J. Chem. Phys. 1990, 93, 4003. In the latter paper, the authors employ the path integral-based theory of refs 19 and 20. However, as opposed to the present work, they take a different approach based on the diabatic 'energy gap" reaction coordinate. (d) Warshel, A.; Hwang, J.-K. J. Chem. Phys. 1986,84, 4938; Science 1989, 246, 1 1 2. (17) Shida, N.; Barbara, P. F.; Almlof, J. J . Chem. Phys. 1991, 94, 3633.
0022-3654/91/2095-10425.$02.50/0
contribution to the activation free energy may come from solvent polarization fluctuations. These fluctuations create the necessary degeneracy between the two solvated proton states and thus allow the proton to transfer between binding sites. (This is, of course, just the usual picture of charge transfer in polar media.18) This picture is to be contrasted with a different viewpoint' which suggests that the activation energy comes from the actual intramolecular potential energy barrier along the proton-transfer coordinate. To-further complicate matters, several authors have pointed out that intramolecular vibrations may also play a crucial role in modulating the proton t ~ n n e l i n g . ' ~ ' ~This ~ ' ~latter ~~'~ behavior comes about because the relatively large mass of the proton causes the exponential tail of its wave function to fall off very rapidly with distance (as opposed to, e.g., that for an electron). As a result, specific intramolecular fluctuations which effectively increase the overlap between the wave functions in the stable binding sites may have a substantial influence on the rate. It therefore seems evident that the interesting interplay between tunneling, solvent fluctuations, and intramolecular vibrations makes proton transfer a very fertile problem for theoretical In the present paper, a Feynman path integral quantum activated rate t h e ~ r y ' ~is* *further ~ analyzed to demonstrate its applicability to proton-transfer reactions. One immediate benefit of this method is that proton transfer can be studied over a wide range of temperatures within a single computational framework. Specifically, the low-temperature (or high proton barrier) d e e p tunneling regime14J5 can be analyzed on an equal footing with the high-temperature limit where the proton itself becomes classically activated. A particularly important feature of the path integral theory outlined in the following sections is that the actual proton-transfer coordinate is employed as the reaction coordinate. At the same time, however, it will be shown that the activated nature of the solvent polarization fluctuations are also accounted for in the quantum free energetics. An alternative, but related, path integral viewpoint'" involves the more standard "energy gap" reaction coordinate consisting primarily of collective solvent modes. A simplified path integral equation is also identified in the present paper by treating the polar solvent as a dielectric continuum. This simpler formulation of the path integral theory, therefore, allows one to efficiently study the role of the intramolecular motions and their couplings to the proton-transfer coordinate. However, it is important to note that (with additional computational effort) an explicit microscopic model for the solvent (18) Ulstrup, J. Charge Transfer Processes in Condensed Media, Lecture Notes in Chemistry; Springer: Berlin, 1979; Vol. 10. (19) Voth, G. A.; Chandler, D.; Miller, W. H. J . Chem. Phys. 1989, 91, 7749. This work also draws on the important insights contained in ref 20. (20) Gillan, M. J. J . Phys. C 1987, 20, 3621. A detailed discussion of the differences between the formula proprosed by Gillan and by the authors of ref 19 (Le., eq 1 of the text) appears in Appendix A of ref 21.
0 1991 American Chemical Society
10426 The Journal of Physical Chemistry, Vol. 95, No. 25, 1991
can be implemented in the numerical studies instead of the dielectric continuum approximation. The latter line of research, however, will be the subject of a future publication. The sections of the present paper are organized as follows: In section 11, the basic path integral equations are outlined along with a discussion of the relevant approximations and different temperature limits. Section 111 then includes an extension of these ideas to the aforementioned simplified model in order to efficiently probe the intramolecular features of many proton transfers. Section IV includes results from calculations on two model systems, and concluding remarks are given in section V. 11. Path Integral Equations The method developed in the present paper is based on a quantum path integral activated rate In that theory, the exact forward rate constant for an activated rate process is written as
k = K-
Li and Voth of quantum treatment is surely indicated (for some alternative treatments of the highly quantum mechanical regime, see refs 14, 15, and 16c). To demonstrate that eq 1 is applicable to “normal” proton-transfer reactions at all temperatures, a phenomenological path integral action can be studied in eq 3 to highlight the salient features of the problem. By definition, the intramolecular proton reaction coordinate is described by a double-well potential VJq) on which the reactive transition occurs (e.g., along an intramolecular asymmetric stretch coordinate involving the proton). For the present discussion, this potential is assumed to be symmetric with minima at (-dw,dw)and to have a maximum at q* = 0 (this assumption is not essential). In turn, the charge distribution of the solute is assumed to be coupled to a collection of effectiue bath harmonic oscillators that represent a linearly responding solvent polarization field. As in the usual picture (see, e.g., ref 15 and references cited therein), the effective coordinates have a mean value of zero are characterized by the “bath” Hamiltonian (where pi = mixi):
kB T exp(-OE*,) hQ*
where QA is the quantum reactant partition function, K is a quantum dynamical correction factor, which has a value of order unity, and the quantum free energy of activation is given by
F*, = -kBT In [ Q * / ( m / 2 ~ h ~ @ ) l / ~ ] ( 2 ) In eq 2, Q* is the equilibrium path centroid d e n ~ i t y , ’ ~defined -~l as the constrained imaginary time Feynman path integral22
The physical characteristics of the effective solvent oscillators can be estimated from dielectric continuum theoryI5 or directly calculated from classical molecular dynamics If the reaction and the solvent coordinates are assumed to interact through a linearized dipole interaction term, the total path integral action in eq 3 is given by
Q* = 1 . . . 1 W 7 ) W 7 ) Nq* - 40) e x ~ W [ q ( ~ ) , x ( 7 ) l / h l
N
i= 1
The constants ci in eq 6 depend on the change in the solute electronic charge distribution when the proton transfers between the two binding sites. The centroid density in eq 3 can be simplified for this proton-transfer model by explicitly performing the path integration over the solvent The result is given by
Q* (4) A “transition state” approximation based on eq 1 can be readily
d7 l(m/2)4(7)2+
Vr[q(~)l+ Hs[X(7)~(7)1+ d 7 ) C ~ i x i ( ~(6) )I
(3)
In this equation, q(7) is the reaction coordinate which has a transition state value of q*, the vector x(7) contains the nonreactive “bath” degrees of freedom, and S[~(T),X(T)] is the imaginary time action functional.22 The quantity q0 in eq 3 is defined as the centroid of the reaction coordinate quantum path q(7) and is given bY‘9-21
s,
hb
S[q(7),x(7)1 =
=
QSS-.JWT) Nq* - 40) ~
X P W J ~ ~ ) I(7)/ ~ I
where Qsis the partition function for the solvent uncoupled to the reaction coordinate, and the effective reaction coordinate path integral action functional Sq[q(7)]is given by
employed in which K is set equal to unity. Such an approach seems to be quite powerful for the treatment of complex problems because only quantum statistical mechanical information is required in eq 1 to estimate the quantum rate ~ ~ n ~ t a n t . ~ ~ ~ J ~ - ~ ~ , ~ ~ , ~ ~ Furthermore, the centroid density in eq 2 can be calculated for complicated potential energy functions by discretized path integral In eq 8, the quantity ~~(1.1) is the kernel of the imaginary time Monte Carlo technique^.^^^^^ influence f ~ n c t i o n a l , ~given ~ , ~ by ~,~~ The quantum rate theory embodied in eqs 1-4 will be shown below to be useful for studying proton-transfer reactions. Indeed, 4171) = (1 / 2 ~L )m d wJ ( 0 )exp(-wl71) (9 1 since such reactions can have large quantum effects, some form (21) Voth, G. A,; O’Gorman, E. V. J . Chem. Phys. 1991, 94, 7342. (22) Feynman, R. P. SfofisticalMechanics; Addison-Wesley: Reading, MA, 1972; Chapter 3. (23) Bader, J. S.; Kuharski, R. A,; Chandler, D. J . Chem. Phys. 1990, 93, 230. A related low temperature, ‘deep tunneling” path integral method to study the Golden Rule rate constant expression has been developed: Wolynes, P. G. J . Chem. Phys. 1987,87,6559. And applied: Zheng, C.; McCammon, J . A.; Wolynes, P. G. Proc. Natl. Acad. Sci. U.S.A.1989,86, 6441. The latter method becomes essentially equivalent in the deep-tunneling limit to the centroid density approach of ref 19. (24) The path integral centroid method of ref 19 has also been implemented to study H-atom diffusion on metal surfaces: Sun, Y.-C.; Voth, G. A., manuscript in preparation. (25) For reviews, see: (a) Berne, B. J.; Thirumalai, D. Annu. Reu. Phys. Chem. 1987,37,401. (b) Freeman, D. L.; Doll, J. D. Adu. Chem. Phys. 1988, 708,139. (c) Doll, J . D.; Freeman, D. L. Ibid. 1989, 73, 289. (d) Quantum Simulafions of Condensed Marter Phenomena; Doll, J . D., Gubernatis, J. E., Eds.; World Scientific: Singapore, 1990. ( e ) Chandler, D. In Liquides, Cristallisafionet Transirion Vitreuse, Les Houches, Session L1; Levesque, D.; Hansen, J. P., Zinn-Justin, J., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1990.
In turn, the influence functional kernel in eq 9 is related to the spectral density J ( w ) of the effective solvent oscillators and is given by the expression28
The continuum limit of this expression, which will be discussed below, will be employed in realistic calculations. The solvent spectral density in the above model is also related to the classical autocorrelation function of the solvent-induced ~
(26) Kuharski, R. A,; Bader, J . S . ; Chandler, D.; Sprik, M.; Klein, M. L.; Impey, R. W. J . Chem. Phys. 1988, 89, 3248. Zichi, D. A.; Ciccotti, G.; Hynes, J. T.; Ferrario, M. J . Phys. Chem. 1989, 93, 6261. (27) Feynman, R. P.; Vernon, F. L. Ann. Phys. (N.Y.) 1963, 24, 118. (28) Calderia, A. 0.;Leggett, A. J . Ann. Phys. (N.Y.)1983, 149, 374; 1984, 153, 445(E). Leggett, A. J.; Chakravarty, S.; Dorsey, A. T.; Fisher, M. P. A.; Garg, A.; Zwerger, W. Rev. Mod. Phys. 1987, 59, 1. Zwanzig, R. J . S f a t . Phys. 1973, 9, 215.
The Journal of Physical Chemistry, Vole95, No. 25, 1991 10427
PT Reactions in Polar Solvents force fluctuations 6Fext(t)on the reaction coordinate at the transition state. This relationship is given explicitly by
where ( ...)q. denotes classical Boltzmann averaging over bath variables which are equilibrated for the transition state (Le., with q = q*). For a realistic microscopic model of the solvent, therefore, the spectral density can be obtained from eq 1 1 by an inverse cosine transform of molecular dynamics data. The central quantity in the quantum rate constant expression (eq 1) is the ratio of the reaction coordinate centroid density Q* to the reactant partition function QA. By employment of the phenomenological action in eq 6, it will now be demonstrated that this ratio of quantum statistical mechanical objects captures the essential features of proton transfer in polar solvents at both high and low temperatures. Furthermore, it will be shown that the activation factor arising from the solvent appears in the present equations even though the actual proton-transfer coordinate is taken to be the reaction coordinate. The required analytical treatment of the path integral in eq (6) is best accomplished by Fourier path i n t e g r a t i ~ n . ~Thus, ~,~~ the imaginary time quantum paths q(7) are expressed as a Fourier series: q(7) =
C 4, exp(iQ,T)
n=-m
(12)
where Q-,,equals 4*,,, and Q, equals 27rnlhp. The effective reaction coordinate action functional in eq 6 is written in the Fourier mode representation as Sq[qnl/ h =
l J h S d r Vr[q(7)]- pCOqo2+ B e ( m Q . 2- 2Cn)14,J2 ( 1 3 ) h o n= 1 where the integral of V,[q(7)] over 7 in eq 13 is understood to be a functional of the Fourier modes g,,, and the quantities Cn are ), by the Fourier transforms of L U ( ~ T ~given Cn p 6(Q,)= 1 -- md .
~~(171) COS ( Q n 7 )
(14)
For the double well potential along the proton-transfer coordinate, the reactant wells are characterized by minimum values at qa = *d, and by a well frequency w,. The high-temperature limit in the proton-transfer problem is characterized by the situation hpw, > 1 and OA
