J . Phys. Chem. 1993,97, 2209-2220
2209
Femtosecond Real-Time Probing of Reactions. 10. Reaction Times and Model Potentialst Qianli Liu and Ahmed H.Zewail' Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125 Received: November 10, 1992; In Final Form: December 8, 1992
In this article, we consider the femtosecond dynamics on model potentials describing antibonding (or nonbonding) and bonding systems. To obtain an analytical description of the dynamics, reaction times are defined with the help of a simple model. Applications to unimolecular and bimolecular reactions are discussed.
I. Introduction In the formulation of transition-state theory by Eyringl and Evans and Polanyi,z the equilibrium assumption3gives rise to the known expression for the macroscopic rate constant
where k is the Boltzmann constant, h is Planck's constant, and Q*and Q are the partition functions of the transition state and the reactants (less the one for the reaction coordinate), respectively. The rate constant is effectively a product of a probability and a "frequency", or inverse "lifetime", factor. In this case, the probability is determined by the energetics and can be expressed as e-EJkT,where E, is some energy barrier and Tis the relevant temperature. Accordingly, assumng Q* Q, the lifetime can be estimated from h/kT which translates to a value of the order of 10-13 s at room temperature. In a microscopic description of any elementary reaction, one must cdhsider the dynamics of the passage of the system through the transition-state and the nature of the internuclear forcesacting during the transition. The degree of localization and the time scales involved were theoreticallyconsidered almost 30 years ago. For example, J o h n ~ t o nbuilding ,~ on the work on potential energy surfaces by Polanyi and E ~ r i n g ,estimated ~ the de Broglie wavelength for H + Hz andother reactions at finite temperatures. For the H + Hz reaction, Karplus6has considered trajectories of the motion and computeda transition lifetimeof 10fs. Polanyi' has provided general descriptions of those features of the intermolecular potential which are important in determining the outcome of a reactive trajectory through the transition-state region, and Levines has considered the nature of the vibrational dynamics and energy disposal. Experimental studies of molecular reactive scatterings have provided a methodology for estimating the lifetime of collision complexes (bimolecular reactions) and half-collision complexes (unimolecular reactions). The key advance was the introduction of the correlationbetween the macroscopic scatteringdistributions and the microscopic rotational period Tr of the complex. Herschbach9 has classified the dynamics according to two time scales. If the complex lifetime, T*,is much shorter than an average T,, then the reaction is said to be direct and the scattering distribution is generally expected to be "asymmetric". If 7* is longer than the average T,,then the reaction is said to be complex modeand thescattering distribution is expected to be'symmetric". In this latter case, T * cannot be unambiguously estimated. Herschbach's osculating complex model is used for the case when 7* Trm9 Lee,loin hiscrossed-beamstudiesof different reactions,
-
-
-
' This article is dedicated to Dudley Herschbach, a brilliant scientist and a great colleague-we wish him the best and expect the Herschbachian contributions to continue with every femtosecond full of joy! OO22-3654/93f 2Q91-22Q9$04.Q0f Q
has shown the importance of scattering experiments for the study of potentialenergy surfaces(relevantto T*),particularlyfor directmode reactions. Reactions of direct/complex mode, or of Feshbach-typeresonance' I have been studied.8 For half-collision reactions,the concept of clocking by the average rotational period is also applicable, as shown by Zare, Bersohn, Wilson, Simons, and others.12-15 But in this case the initial state of the reaction is more well-defined. With femtosecond resolution, it is possible to resolve in realtime the passage through the transition state and to determine thereaction time (7)or the transition-statelifetime (7*).16 Studies of unimolecular and bimolecular reactions have been carried out by establishing the zero-of-time with this resolution. Classical, quantum mechanical, and semiclassical calculations have been s u ~ c e s s f u in l ~ describing ~ major features of these experimental results for different typesof reactions. Although thesecalculations are important in providing quantitative comparisons between theory and experiment,it is helpful to develop some simplemodels that allow one to consider key parameters which influence the dynamics. In this paper, we invoke the energy conservation equation 1 E =p ~ * V(R)
+
(1-2)
to describe the nuclear motion of the wave packet on different potential energy surfaces (PES'S). With the help of the relationship AE T[T*] h,Is we deduce analytical expressions for 7 (or 7*)in terms of the force and other characteristics of the potential. The wave packet dispersion on different PES'S is also obtained in order to find the time scales on which a particle-like (short de Broglie wavelength) description of nuclear motions is valid. Applicationsare made to femtosecond dynamics on bound potentials (harmonic and Morse type) and on repulsive or saddlepoint type potentials. In a later publication, we will consider recent experimental studies of unimolecular and bimolecular reactions, compare results with those derived from quantum and other types of calculations, and include the influence of finite pulse widths.
-
11. Unimolecular and Bimolecular Reactions
The reaction complexes that we shall consider are generally of two different types, resulting from unimolecular and bimolecular reactions: ABC A
+h
+ BC
~ [A*B*C]* + -AB
-
[A-BC]' -AB
+C +C
(11-1)
(11-2)
For thecaseof (11-l), theinitialvalueRioftheseparationbetween AB and C in [A.B.C] * is related to the total available energy E, as shown in Figure 1 in the center-of-mass (CM) frame. Taking Rfas the point at which the dissociation is over, we have the 0 1993 American Chemical Society
Liu and Zewail
2210 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
I
I
I
1
I \
Exponential Potential
Figure 1. Repulsive potential energy curve: V(R)= Ee-(n-n,)/L.A wave packet is shown to represent the starting point of the dissociatingsystem. The parameters shown are defined in the text.
following classical expression for the bond-breakage time 7
=S
R
R
'
e
RO R Figure 2. Schematic of a typical effective potential energy curve Y,rr(R) ofthe form V ( R )+ 1(1+ 1)ft2/(2&) for a bimolecular reaction leading to a complex formation. A wave packet represents the collision of two particles. Ro corresponds to the position of maximum barrier height along the collision coordinate.
(11-3a)
where E is the initial energy of the system above the dissociation limit, and u(R) = ( 2 [ E - V(R)])'/2 (11-3b) P is the classical momentary velocity at separation R. V(R) is the correspondingpotential energy and p is the reduced mass of the separating nuclei. Bersohn and ZewailI9have given a definition of T * by considering the absorption of separating fragments at different internuclear distances and a definition of T when the separation reaches the terminal Rfvalue of free fragments. Also, using classical mechanics, Bernstein and Zewai120 have used similar ideas to invert measurements of T and T * to the PES.For example, for the dissociation process of ICN, with V(R) = Ee-(R-Rl)/L, the time given by eq 11-3a is about 200 fsI6if L = 0.8 A and the initial energy E is 6000 cm-I. In this case, Rr - Ri is -4 A. From a pure classical mechanics point of view, the assignment of the final position Rr is quite arbitrary, but, as discussed below, a simple model can relate Rr to the time scale of interest. For bimolecular reactions, the overall process can sometimes be treated as combinations of unimolecular steps. In the case when two molecules come close together to form a complex, the process can be linked to a time-reversed unimolecular dissociation. It is, however, more difficult to deal with bimolecular reactions because the initial conditions are not as well-defined at the microscopic level. Molecules could collide with each other at different velocities and different angular momenta, both of which are important factors for the outcome of collisions. In terms of PES,one must include the centrifugal term L 2 / 2 p R 2(or 1(1 + l)h2/2pR2in quantum mechanics) in the discussion of collision dynamics, unless the angular momentum L is zero. Therefore, an effective potential, V,rr(R), is employed
with V,rr(R) being different depending on the collision geometry. As shown in Figure 2, this centrifugal term gives rise to an energy barrier at the entrance channel over which colliding molecules must pass for reactions to take place. If the colliding molecules do not have enough kinetic energy to overcome this barrier, they will approach only as close as some distance located 'outside" the potential barrier before separatingagain. Their internal electronic states will not be significantly perturbed during this process and the collision will not lead to a chemical reaction.
1
a) Direct-Mode Bimolecular Reaction
I
b) Complex-Mode Bimolecular Reaction Figure 3. (a) Schematicillustrationof a direct-mode bimolecularreaction in the CM frame. 60is the critical impact parameter as defined in the text, and Ro is the corresponding orbiting distance. (b) Schematic illustration of a complex-mode bimolecular reaction in the CM frame.
The angular momentum can be expressed as L = bpu (or (l(1
+ 1))'/2h in quantum mechanics), where u is the initial relative
velocity of the colliding molecules. The impact parameter b is defined as the closest distance of approach between the two molecules in the absence of any interaction between them. The collision geometry is shown in Figure 3a for a direct-mode reaction. If it is assumed that the colliding molecules will not react unless they can overcomea centrifugal barrier, then a reaction will occur only for those collisions with impact parameters smaller than a critical value bo. This value of bo is the solution to the following equations (11-5) (11-6) where Ro is the position of the barrier maximum as shown in Figure 2. The above equations describe the situation in which
Femtosecond Real-Time Probing of Reactions
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2211
the initial energy of collision E = pu2/2 and the impact parameter bo are such that the centrifugal and attractive forces balance each other and the two colliding particles orbit around each other at a distance of Ro. For the same initial energy, if b < bo,the centrifugal force will fail to stop the colliding pair from reaching separations smaller than Ro and undergoing reaction. If b > bo, on the other hand, the colliding pair will not approach closer than Ro. In terms of reactive scattering experiments,d o 2 corresponds to the upper limit of the reactive scattering cross section. The orbital period around Ro can be expressed as
-.-I
2
\!>
11%-
j
.V....W...).
1
i
j
j
j
............... ............ i............................
b...
...............................
.......................
j
0
j
~
(11-7) wherelis the moment of inertia of thecomplex at Ro. The period T,has been used as a ‘clock” for the microscopiccollision complex, assuming this value of L for a given geometry. The angular distribution of reactive scatteringmay be related to the microscopic rotational angle 8, of the collision complex,and the time scale can be estimated from (11-8) If the lifetime of the collision complex is so short that its dissociation is completebefore the complex rotates an appreciable angle, then the complex should still “remember” the initial geometry when it breaks apart. In other words, the ensemble of the reactive complexes behaves more or less coherently if their average lifetime is much shorter than the averagerotational period; there is not sufficient time for differences in their angular momenta to average out the inherent dynamical characteristics of the reactive process. The reactive scattering distribution in this case is asymmetric in the center-of-mass system. Conversely, based on the asymmetry of the distribution, one may infer that the lifetime is shorter than the microscopic T,defined in (11-7). When the lifetime of the complex is, on average, longer than or many times the average rotational period, the scattering distribution, which is averaged over all the different angular momentum distributions, will no longer be able to provide an estimate of T * of the complex. It is possible to study bimolecular reactions with relatively well-defined initial conditions. If the two molecules are initially confined in a van der Waals complex, the range of the impact parameters will be limited, as demonstrated elegantly by Wittigzl and by S ~ e p For . ~ this ~ case of bimolecular encounter, the zeroof-time can be established and the reaction dynamics can be resolved in real time.23 The centrifugal effect was not mentioned ‘when discussing unimolecular reactions. The centrifugal term L2/2pR2 basically adds a repulsive contribution to V(R). For dissociation, it makes the whole process somewhat faster depending on the magnitude of the orbital angular momentum L. In contrast to bimolecular reactions, small impact parameters (smaller centrifugal effect) are expected for unimolecular dissociations. However, if significant, the centrifugal term, which decreases with R-2 if L is conserved, can be included. The velocity is then expressed as
u(R)
(:)I/’[
E - V(R) -
where 1 is the quantum number of the orbital angular momentum of the dissociating system. The total angular momentum of the initial complex is J = L + j, where j is the angular momentum of the dissociating fragments and L is the orbital angular momentum. 111. Reaction Times
We now consider two potential energy curves in a system such as shown in Figure 4: curve X represents a bound ground state;
R Rr R Fipre4. Relevant potentialenergy surfaces in a unimoleculardissociation process: curve X represents the ground state and curve A represents the upper dissociative state. Two wave packets are drawn to illustrate the photon (with energy hw) excitation process. The parameters shown are defined in the text. the other, labeled A, is an upper state which could be either nonbonding or antibonding (repulsive) as shown. Suppose that the system (e.g. an ABC molecule) is initially vibrating near the minimum of the X state. At a certain time a photon of frequency w is absorbed. As a complex of the type [A-BC]* is generated in a non- or antibonding state, the system reorganizes to lower the potential energy and form AB + C. The classical equation (11-3a) can be used to describe the dissociation process as if AB and C are particles with mutual repulsion. According to eq II-3a, the bond-breakage time is determined by the range of the integration from Ri to Rr. As mentioned before, in strict classical treatment Rr may go to infinity and accordingly T will also reach infinity. However, here we shall consider Rr to be consistent with the measurement time in order to obtain analytical estimates of T and T * . In this way Rr and the terminal value of the potential Vr can be related. Equation 11-3a can be rewritten as
where, ‘‘..,’’ stands for the parameters used to describe V(R), and Vr is the most probable final potential energy at which the nuclei are indistinguishable from the situation in which the interaction between them is zero, as far as the measurement of bond-breaking time is concerned. Quantum mechanically,for unstable systems, the motion of a wave packet up to certain time r leads to a correspondingvalue AE.18 Equivalently, the precision by which the energy of the system (and the potential energy) is known depends on the time over which the nuclei effectively separate. To be consistent with the time scale of the process, we shall approximate Vr by AE and hence relate it to T by h. For the system in transition one may introduce similar terminology. The system passes through a transitional configuration defined by the internuclear separation R*, and AV(R*) is defined as the energy “window” around R*.19920hR* is the spatial range correspondingto AV(R*) and is related to AV(R*) by the following equation:
Taking the leadng term in the expansion, one obtains
For other problems of interest (e.g., the case of a saddle-point treated below), the leading term could be quadratic in hR*. Classically, the transient time T * for the system to stay within
Liu and Zewail
2212 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
1
I
this AR* can be written as
Half-Harmonic Potential
(111-3a)
i
;
Assuming that AR* is small enough, we can approximate the above integration by the following expression (111-3b) where
u(R*)= ( 2 w - 7 R * ) 1 ) ’ / 2
(111-4)
is the most probable velocity of the nuclei around the separation R*. Combining (111-2b) with (111-3b), we obtain (111-5) whereF(R*) corresponds to the classical force at R*. According totheaboveequation, thenarrower theenergy window theshorter the transient time. It seems that the transient time could potentially go to zero as AV(R*)goes to zero. However, a wave packet of finite width in energy will have a corresponding spatial spread AR, and this “preventsn the AV(R*) from going to zero. If we take the energy width of the wave packet to be comparable to AV(R*), T * may be related to AV(R*) by h. In this limit
Ri R’ Rf Ro R Figure 5. The solid line represents a special type of repulsive potential,
which may be termed a half-harmonic potential. It is constructed by taking only the left half of a harmonic well and attaching to it from the minimum point Ro a straight line. The dotted line shown is a mirror reflection of this potential with respect to Ro. The other parameters shown are defined in the text.
IV. The Potentials A. Harmonic Potentials. We first consider a dissociative potential which is the “left half“ of a harmonic potential well (Figure 5 )
V(R)=
j p o 2 ( R- Ro)2 ...(R IR,) ...(R
AV(R*) = [ hu(R*)IF(R*)l]‘ I 2
(111-6)
and (111-7)
Assuming that the initial most probable position is Ri, where the system has a most probable velocity of zero with a total energy E, the system in time T will move, on the average, to Rr where the potential energy is most probably measured as Vr. Then R , - R , = ( R i- R,)
Substituting (111-6) into (111-2b), it follows that (111-8) We will refer to T * as the transition-state lifetime because it indced describes the average time that the system spends around a certain internuclear distance. The separation R* now becomes the most probablevalue which is intrinsic in the nature of measurements, if we repeat the measurements enough times on identical molecular systems, one can find the most probablevalue of the potential energy measured around R*. The most probable value describes the potential energy which governs the motion of the center of the wave packet. In section VI, we shall consider the extent of the spreading of the wave packet. In the followingsection, reactions on different types of potentials are considered, and analytical expressions for both the dissociation time 7 and the transition-state lifetime T * are derived. Here, for simplicity, we shall not deal with the actual shape of the wave packet (which incorporates the shapeof the preparation and probe pulses). It should be noted that for bound or quasi-bound systems, the preparation and probing are straightforward processes to understand, even classically, as the time scale of the dynamics16 is longer than the pulse widths. If the dynamics is on a very short time scale, as for purely repulsive potentials, then the shape of the initial wave packet is clearly part of the dynamics, but the slower reaction dynamicsZ4basically reflects the motion of any portion of this packet as shown experimentally16 and theoretially.*^.^^ Mukamel has examined this point classically and quantum mechanically for arbitrary pulse duration.25
(IV-A-1)
> R,)
COS WT
(IV-A-2)
where w would be the characteristic frequency of the harmonic potential well if the right half (the dotted line in Figure 5 ) were included. Using the relations
and (IV-A-3b) we obtain ( V f / E ) ” 2= COS UT
(IV-A-4)
Now, for convenience, we substitute Vf by (h/r)(x/4) in the above equation and obtain ( w T ) ’ / ~COS U T
= ( ~ h o / 4 E ) ’ / ~ (IV-A-5)
We can write E = (n + 1/2)hw, where n would be the vibrational level corresponding to the excitation energy E, if the system were in a harmonic well. Equation (IV-A-5) can be rearranged as COS W T
= C/(WT)’’2
(IV-A-6a)
with (IV-A-6b) There are generally two solutions to W T in the range from 0 to ?r for n larger than 1, as shown in Figure 6. The solution that
Femtosecond Real-Time Probing of Reactions
The Journal o/ Physical Chemistry, Vol. 97, No. 10, 1993 2213 interest. Equation IV-A-11 shows that, in the classical limit: n > n* >> 1,7*is much smaller than I as expected. For example, at the point of n* = n/2, T* is l/w(n2 n)Il4, which is less than T by a factor of 2/r(n2 + n)1/4when n is very large. B. Exponential Potentials. We now turn to a more realistic potential for repulsion of the type denoted A in Figure 4:
+
V(R) = Ee-(R-Ri)/L
(IV-B- 1) where E is the total available energy. For this potential, the force is given by
Figure 6. Schematic of the mathematical solution to eq IV-Ad: cos u T = c / ( w T ) ' / * . The arrow points to the selected solution of interest.
TABLE I: Numerical Solutiona to Equation IV-A-5: eo9 w = [r/2(2lJ + 1 p n=+n=100 n=20 n-10 n=4 WT
r/2
0 . 9 6 ~ 1 2 0 . 8 9 ~ 1 2 0 . 8 5 ~ 1 2 0.15r/2
is closer to 1/2, which corresponds to Rr
-
According to eq 111-1, we can now obtain the bond-breakage time
(w)l/*
n=2 0.61~12
Ro,is selected. Let
U
(IV-A-7)
W7=2--Y
In cases when n >> 1 or c O
Schematicof the solution to eq IV-B-6c: In y = cy. The arrow drawn points to the selected solution of interest (see text). Figure 7.
TABLE II: Numerical Solutions to Equations IV-B-6 y = 4 E / h c = h/2L(2pJO1/2; In y = cy, and T = (ti/4@p L (A) C
Y
Vf(cm-I) 7 (fs)
0.3 0.0 19 30 1 80 61
0.5 0.01 1 556 43 123
0.8 0.007 1 967 25 214
Parameters used are E = 6000 cm-I and
fi
1 .o 0.0057 1255 19 278
= 21.6 amu.
depending on the value of L (Table 11). For this system, when, e.g., L = 0.8 A, we obtain Ri - Ri = 4.4 A and Vf = 25 cm-I > 1.
References and Notes ( I ) Eyring, H. J . Chem. Phys. 1935, 3, 107. (2) Evans, M. G.; Polanyi, M. Trans. Faraday SOC.1935, 31, 875. (3) Laidler, K. J.; King, M. C. J . Phys. Chem. 1983, 87, 2657. (4) Johnston, H. S.Gas Phase Reaction Rate Theory;The Ronald Press: New York, 1966. (5) Eyring, H.; Polanyi, M. J . Phys. Chem. 1931, 12, 279. ( 6 ) Karplus, M.; Porter, R. N.; Sharma, R. D. J . Chem. Phys. 1%5,43, 3259. (7) Polanyi, J. C. The Chemical Bond Structure & Dynamics; Zewail, A. H., Ed.; Academic Press: Boston, MA, 1992, and references therein. (8) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemica1Reactiuity;Oxford University Press: New York, 1987,and references therein. (9) Herschbach,D. R. Reference7;seealsohisreview,ReacriueScartering in Molecular Beams. Adu. Chem. Phys. 1966, 10, 319. (10) Lee, Y. T. Science 1987, 236, 793. (1 I ) Miller, W. H. Annu. Reu. Phys. Chem. 1991,41,245,and references therein. (12) Greene, C. H.; Zare, R. N. Annu. Reo. Phys. Chem. 1982.33, 119. (13) Dmonik, M.; Yang, S.;Bersohn, R. J . Chem. Phys. 1974,61,4408. (14) Riley, S.J.; Wilson, K. R. Faraday Discuss. Chem. Soc. 1972, 53, 132. (15) Simon, J. P. Molecular Photodissociation Dynamics; Ashfold, M. N. R., Baggott, J. E., Eds.; Royal Society of Chemistry: Hertz, 1987. (16) Zewail, A. H. Faraday Discuss. Chem. Soc. 1991, 91, 207, and references therein. (17) See the theoretical papers in Faraday Discuss. Chem. Soc. 1991,91, and references therein. (18) See, e.&, Satchler, G.R. Introduction to Nuclear Reactions, 2nd ed.; Oxford University Press: New York, 1990. For the proof of the generality of hE At 2 h / 2 in relation to the fundamental Ap Ax 2 h/2,see Mandelstam, L.; Tamm, Ig. J. Phys. USSR 1945, 9, 249. (19) Bersohn, R.; Zewail, A. H. Ber. Bunsen-Ges. Phys. Chem. 1988,92, 373. (20) Bernstein, R. B.; Zewail, A. H. J . Chem. Phys. 1989, 90, 829. (21) Shim, S.K.;Chen, Y.; Bohmer, C.; Wittig, E. Dye Losers: 25 Years; Stiike, M., Ed.;Springer-Verlag: New York, 1992;p57,and references therein. (22) Duval, M. C.; Soep, B.; Breckinridge, W. H. J . Chem. Phys. 1991. \ 95, 7145, and references therein. (23) Sims, I. R.;Gruebele, M.; Potter, E. D.; Zewail, A. H.J. Chem. Phys. 1992, 97, 4127, and references therein. (24) Beswick, J. A.; Jortner, J. Chem. Phys. Lett. 1990, 168, 246. (25) Fried, L. E.; Mukamel, S . J . Chem. Phys. 1990,93, 3063. Yan, Y. J.; Fried, L. E.; Mukamel, S.J . Phys. Chem. 1989, 93, 8149. (26) Roberts, G.; Zewail, A. H. J . Phys. Chem. 1991, 95, 7973. (27) See, for example, Schiff, L. I. Quantum Mechanics, 3rd ed.; McGraw-Hill: New York, 1968; pp 74-76. (28) Krim, L.; Qiu, P.; Jouvet, C.; Lardeux-Dedonder, C.; Mccaffrey, J. G.;Soep, B.; Solgadi, D.; Benoist,d'Azy, 0.;Ceraolo, P.; Hung, N. D.; Martin, M.; Meyer, Y. Chem. Phys. Lett. 1992, 200, 267. (29) Hynes, J. T. In The Theory ofchemical Reaction Dynamics; Baer, M.,Ed.;CRC Press: Boca Raton, FL, 1985; Vol. 4, p 171. (30) Schatz, G. C.; Dyck, J. Chem. Phys. Lett. 1992, 188, 11. (31) Scherer, N.;Sipes,C.; Bernstein, R. B.; Zewail, A. H. J . Chem.Phys. 1990, 92, 5239, and references therein. (32) lonov, S. 1.; Brucker, G. A,; Jaques, C.; Valachovic, L.; Wittig, C. J . Chem. Phys., in press. (33) Skodje, R., private communication, work to be submitted for publication. (34) Williams, S.0.;Imre, D. G. J . Phys. Chem. 1988, 92, 6648. (35) Metiu,H. Faraday Discuss. Chem.Soc. l991,91,249,and references therein. .... .. ....
(36) See,e.&, Cohen-Tannoudji, C.; Diu, B.; Lalo€, F. Quantum Mechanics; Hermann: Paris, 1977; p 64. (37) See Bernstein, R. B.; 2ewail.A. H. Chem. Phys. Leu. 1990,170,321, and references therein. (38) Very recently, we have learned of the work of L. Valko (private communication, to be published) which defines a unique dissociation time on a Morse-type potential.
