J. Phys. Chem. 1991, 95, 10235-10240 Mukai of Ehime University for his continuous encouragement. The presentation of the paper in Kasha Conference by S.N. was supported by a grant from The Ministry of Education, Science and Culture of Japan. At last but not least, S.N. is deeply grateful to Professor Paul F. Barbara of the University of Minnesota for
10235
his kind invitation to submit a paper for publication in this special issue of The Journal of Physical Chemistry in honor of Michael Kasha. Registry No. MS,135952-76-0; OHAP, 125507-95-1.
Variational Transition-State Theory for a Dissipative Cubic Oscillator Eli Pollak Chemical Physics Department, Weizmann Institute of Science, Rehovot, 76100 Israel (Received: February 21, 1991)
New developments in the application of variational transition-state theory to activated rate processes in dissipative media are reported. A variational solution for the optimal dividing surface in configuration space is found. The canonical flux is proportional to the classical action along a classical trajectory evolving under the dynamics of a temperature-dependent 2 degrees of freedom Hamiltonian. This result is of general validity for 2 degrees of freedom systems and so of interest also for thermal reaction rates in conservative systems. An application of variational transition-state theory to a cubic oscillator in the presence of ohmic dissipation is presented. Here, the dividing surface is curved; however, we find that the Krzmers estimate for the rate is valid for almost all parameter regimes.
1. Introduction The theory of decay of a metastable state in a condensed phase remains even today an active area of research. The problem was introduced by Kramers more than 50 years ago.' In his famous paper, he considered the case of a particle in a potential well interacting with a Markovian heat bath. He showed that when the interaction was weak, the rate-determining step was the rate of energy diffusion of the particle. In the moderate to strong damping limit, the rate was limited by a spatial diffusion process. The equation of motion considered by Kramers was a Langevin equation of motion or, equivalently, a Fokker Planck equation in two variables, the coordinate and velocity of the particle. In more recent years, the Kramers problem was generalized to include memory by considering the dynamics defined by a generalized Langevin equation (GLE). A recent excellent review of the Kramers problem and its generalization may be found in ref 2. For the spatial diffusion limit, in which one can assume that the *reactants" are in thermal equilibrium, Grote and Hynes3 obtained an analytic result for the rate (which was later formalized by Hanggi and Mojtabai4). This was a very important step, since it was realized that, especially in a liquid, molecular solvent time scales are of the same order of magnitude as barrier crossing time scales and therefore cannot be ignored. The practical applicability of the GLE to realistic molecular systems has been studied intensively by Berne and co-worker~.~The validity of the GroteHynes expression has by now been tested successfully against detailed molecular dynamics computation^.^^^ This expression is though only a steepest descent estimate for the rate. It is based on a local harmonic expansion of the potential around the top of the barrier. Although expected to be useful for high barriers, it is not always clear whether it is an upper bound or whether it is in fact a good estimate. Recently a variational transition-state theory (VTST) has been formulated for the spatial diffusion limit of the Kramers problem and its generalization to include memory friction.',* This approach ( 1 ) Kramers, H. A. Physicu 1940, 7 , 284.
(2) Hanggi, P.; Talkner, P.; Borkovec, M . Reu. Mod. Phys. 1990,62, 251. (3) Grote, R. F.; Hynes, J . T. J . Chrm. Phys. 1980, 73, 2715. (4) Hanggi, P.; Mojtabai, F. Phys. Reu. A 1982, 26, 1168. (5) Straub, J. E.; Borkovec, M.; Berne, B. J. J . Chem. Phys. 1988, 89, 4833. (6) Gertner, B. J.: Wilson, K. R.; Hynes, J. T. J . Chem. Phys. 1989, 90, 3537. (7) Pollak, E.; Tucker, S. C.; Berne, B. J . Phys. Rev. Lett. 1990,65, 1399.
0022-365419 112095- 10235$02.50/0
is based on a transformation of the generalized Langevin equation to a Hamiltonian in which the system particle (with coordinate q and mass m = 1) is bilinearly coupled to a bath of harmonic oscillators. The VTST expression provides a rigorous upper bound to the decay rate.9 With little loss of generality, the system potential may be decomposed into two parts
where u* is the barrier frequency and V l ( q )is the anharmonic part of the potential. The VTST was used in ref 7 to prove that if V , ( q )is positive definite, then the Grote-Hynes estimate for the rate is in fact an upper bound. The purpose of the present paper is to further develop the variational theory and to present an application for the cubic oscillator. In a previous paper* only a partial solution to the variational problem was described. Here we show in section 2 that, for a dividing surface in configuration space, the full variational solution for the canonical rate is composed of classical trajectories which evolve under the classical equations of motion of a temperature-dependent, effectively 2 degrees of freedom Hamiltonian. This result is actually not specific to dissipative systems. It is of general validity for any 2 degrees of freedom Hamiltonian system. The second part of this paper provides an application of VTST to a system for which the curvature of the dividing surface is important. Perhaps the simplest model for such a case is the cubic oscillator
which is also of practical interest in the theory of Josephson junctions.2,'0 In this case, the anharmonic part of the potential (1.3) attains negative and positive values so that the Grote-Hynes theory gives only an estimate without any bounding properties. In fact, if one uses the straight line dividing surface implied by the (8) Pollak, E. J . Chem. Phvs. 1990. 93. I 116. (9) Wigner, E. P. Truns. Furuduy Sor. 1938. 34, 29. Kcck. J . C. Adr. Chem. Phys. 1967, 13, 8 5 .
0 1991 American Chemical Societv
10236 The Journal of Physical Chemistry, Vol. 95, No. 25, 1991
Grote-Hynes theory, then VTST gives only the trivial bound a, In section 3 we present a variational result for the dividing surface of the cubic oscillator and use it to derive rigorous upper bounds for the escape rate. It should be stressed that all formal results derived in this section are valid for arbitrary memory friction. A numerical application for (Markovian) ohmic friction is also provided. The paper ends with a brief discussion.
2. VTST for Dissipative Systems a. Preliminaries. The theory is developed for the generalized Langevin equation of motion q + - dV(q)
dq
+ J'dr
y(t -
7)
q(7) = [ ( t )
Pollak
The canonical equilibrium flux F through any dividing surface9J4J5is defined as = -m
(2.7) where p is the generalized velocity vector in phase space
E = P # p + bij
(2.1)
The Gaussian random force E(t) is related to the friction kernel y ( t ) (with = l/kb7') through the second fluctuation dissipation theorem: ( [ ( t ) [ ( O ) ) = (l/P)y(t). The system potential V(q) has a barrier at q = 0 and a well at q = qw with harmonic frequency wg. The energy difference between the barrier and well is the barrier height, denoted V . The Grote-Hynes expression for the rate, based on the parabolic barrier approximation, is
A* 0 0 r = ;WA* r0 = - exp(-pv') w* 27r
dPy, dyJ dPp dp wp)e ( v p > 'Y) exp(-flH)
+ E M y p y J + $$jI
and the carat denotes here the unit vector (not the Laplace transform). The derivative with respect to the time is determined through Hamilton's equations of motion derived from the full Hamiltonian. The rather complicated expression for the flux given in eq 2.7 may be substantially simplified by noting that the main difficulty comes from the anharmonic part of the potential VI which mixes all coordinates nonlinearly. However, this is done only through the sum x j u ~ j It. is therefore useful to define a generalized bath coordinate u and bath momentum pc:
The reactive frequency of Grote and H y n e ~ A*, , ~ is the solution of the equation (2.3)
(2.8)
(2.10) When the dividing surface is a function of only the two variables and u, then it is possible to show with some straightforward manipulation (cf. refs 7 and 8) that the flux integral reduces to the much simpler form p
where ?(s) denotes the Laplace transform of the friction kernel y ( t ) . Equation 2.2 is a well-know generalization of the Kramers spatial diffusion limit to include memory effects expected to be valid and relevant for moderate to strong damping and for large barriers (pv' >> 1). A variational transition-state theory may be derived by studying the dynamics of the equivalent HamiltonianIo."
where the system coordinate q is coupled linearly to a bath of harmonic oscillators with coordinates xJ and frequencies wJ. By solving explicitly for the time dependence of each of the bath coordinates, it is found that Hamilton's equation of motion for the system coordinate q reduces to the GLE (eq 2.1), with the identification that y ( t ) = cJcJ2/w;cos (w,t). For a purely parabolic barrier, the Hamiltonian in eq 2.4 is quadratic and may be diagonalized using a normal-mode transformation.12 The normal modes are characterized by one unstable mode p associated with the negative eigenvalue -A*' and by stable modes yJ with associated frequences A,. The unstable mode frequency A * is identical to the reactive frequency given by eq 2.3. The derivation of a VTST proceeds by rewriting the Hamiltonian in terms of the normal modes (of the saddle point):
H = '/z[Pp2- A*'P2 + Z(Py,'
+ 'J2yJ2)] + Vl(U00P + ~
U J ~ J )
(2.5) The uJ;s are elements of the orthogonal normal-mode transformation such that q = uoop + ZJuJg,.The matrix element uoo is defined in the continuum limit through the Laplace transform of the friction kernel:I3 (IO) Caldeira, A. 0; Leggett. A . J. Phys. Reo. Lett. 1981, 46, 21 I ; Ann. Phys. N . Y . 1983, 149, 374. ( 1 1 ) Zwanzig, R. J . Stat. Phys. 1973, 9, 215. (12) Pollak, E. J. Chem. Phys. 1986,85, 865. Levine, A. M.; Shapiro, M.; Pollak, E. J . Chem. Phys. 1988, 88, 1959.
(2.1 1) where
The reduced Hamiltonian H*is found to be
H*=-+--2 2
2
++ VI(p,u) 2
(2.13)
and finally the collective bath mode frequency i2 is
Ai ',A
At2
w * ~
The last equality on the right-hand side follows from properties of the normal-mode transformation.'J3 The flux integral has thus been reduced from an infinite number of degrees of freedom to a tractable 2 degrees of freedom problem which is well-defined in the continuum limit. All the parameters and frequencies (A*, i2, uo, u I ) have been expressed through eqs 2.3, 2.6, 2.9, and 2.14 in terms of the Laplace transform of the time-dependent friction and parameters of the system potential V(q). Therefore, this theory is well-defined for arbitrary memory friction. Some of the interesting physics of the problem now enter through the choice of the dividing surface f. ~~~
(13) Pollak, E.; Grabert, H.; Hanggi, P. J. Chem. Phys. 1989, 91, 4073. (14) Pechukas, P. In Dynamics of Molecular Collisions, Part B Miller, W. H., Ed.; Plenum Press: New York, 1976; Chapter 6. (15) Truhlar, D. G.; Garrett, B. C. Annu. Reu. Phys. Chem. 1984,35, 159.
The Journal of Physical Chemistry, Vol. 95, No. 25, 1991 10237
Dissipative Cubic Oscillator If the barrier is purely harmonic (VI = 0), then the choicef = p is optimal, giving a simple result for the flux integral: (2.15)
It is though possible, at least formally, to solve the variational problem exactly, this is the main purpose of the present section. Inspection of eq 2.19 shows that the transmission coefficient is just a line integral of a positive function. That is, let the length of the dividing surface be parametrized by s. Since ds2 = du2 d 2 , the transmission coefficient has the general form j d s Z(u,g), where Z is positive. Finding that line along which the integral is minimal is an old problem in the calculus of variations, dating back to Fermat and Jacobi.” The result is that the optimal path is a classical trajectory whose motion is governed by some potential. In the following, we will carry out this variation explicitly to find the Hamiltonian that generates the optimal line. The first variation of P is
+
It has been shown elsewherel2.l6that this expression leads to the Grote-Hynes expression for the rate, demonstrating that eq 2.2 is just the harmonic transition-state theory estimate. From here on, instead of using a cumbersome expression for the flux, we will normalize it with respect to the harmonic flux, thus defining a transmission coefficient P as (2.16) and the rate may be written as
A* r = P,ro
(2.17)
W
With small loss of generality, the dividing surface is chosen to have the form f=P(2.18)
where the notation g’is used for the derivative dg/du. Since 6g’ = dbg/du, the first term on the right-hand side can be integrated by parts. Assuming that at the boundary (u = -0, m) the exponent goes to -OD, we find
Insertion into eqs 2.12 and 2.16 and integration over p , pp,and p n give an explicit expression for the transmission coefficient: P = ( f ) 112 j a d u [ l + dg(u) ‘I2 exp[-g[n2uz-
(z) ]
-a
A * ’ ~ ( u ) ~ ] - PVl[uo,g(~)
+
1
U ~ U ]
(2.19)
This result is central to VTST for dissipative systems. Variation of the dividing surface g(u) will lead to minimization of the transmission coefficient. A trivial choice for the dividing surface for any potential is the choice q = 0 or, in the notation of eq 2.18,f= p - (ul/uoo)u.With this choice, one finds that the transmission coefficient is just P = w*/A* and the rate is identical to the so called transition-state theory rate of Kramers. Although trivial, this choice is, as shall be shown later, important, since it demonstrates that, for any value of the damping, there always exists a dividing surface through which the canonical flux is finite. b. Variational Solution. Thus far, the dividing surface has not been specified. In a previous paper* a partial solution of the variational problem was obtained by maximizing the functional only in the exponent: E[g] = l/Z(QV -A*y)
+ vl(u0og + UlU)
(2.20)
The first variation of the functional leads to the determination of the dividing surface in terms of the anharmonic part of the potential
Allowing for arbitrary variations of 6g leads after some short manipulations to the result:
For small curvature of the dividing surface, this leads to the previous procedure with the result given in eq 2.21. The more general solution of this differential equation becomes transparent if one treats g and u as variables parametrized by a single independent “time” parameter t , so that, for example, g’ = g/u. One then finds that eq 2.25 is equivalent to the representation g - g’u =
[
(g‘+ &2)@ g’-
g2
+ u 2 % 2 [ €- V@(E)]
(16) Dakhnovskii, Yu.; Ovchinnikov, A. A. Phys. Lett. 1985, 113A, 147.
(2.26)
(2.27)
where e is as yet an arbitrary energy and we will define the potential V, such that Hamilton’s equations of motion are obeyed exactly g=--
where Vl’(q) = dVl(q)/dq. The second variation of the functional is
and eq 2.21 is a valid variational solution provided that the second functional derivative is negative. As shall be shown in the next section, this added condition is not always obeyed, leading to some additional constraints on the dividing surface. To obtain analytic results for the transmission Coefficient, we find that eqs 2.21 and 2.22 are useful. A specific application to a cubic potential will be presented in the next section.
-
This result is very suggestive, since one almost sees Hamilton’s equations of motion in this form. To make this more concrete, we demand that
(2.21)
(2.22)
-
av@ ag
(2.28a) (2.28b)
in conjunction with eq 2.26. This leads to a simple equation for the depenedence of the effective potential V, on E
av, _ -- 2@(€- v8) dE which is easily solved.
,- E
I/
oe-28EkPl
+e
(2.29)
(2.30)
(1 7) For some history on this problem, see Chapter 44 in: Landau, L. D.; Lifshitz, E. M. Mechanics, 3rd ed.; Pergamon Press: Oxford, 1976.
10238 The Journal of Physical Chemistry, Vol. 95, No. 25, 1991 Here, E, is an arbitrary constant with the dimensions of energy. Combining eqs 2.27-2.30 leads to the conclusion that the variational dividing surface is the configuration space path of a classical trajectory, which is the solution of Hamilton's equations of motion for the effective, temperature-dependent Hamiltonian
H, = y2(p;
+ )p: + Eoe-2~E[@'1
(2.31)
at the energy H, = 0. It is evident that the magnitude of Eo is of no interest since it only serves to scale the time but does not change the configuration space path of the trajectories. For the sake of convenience, we give it the value -1/2/3. It is instructive to analyze this Hamiltonian and its dynamics in limiting cases. For the purely parabolic potential (VI = 0), the initial condition g = 0, g = 0 leads to a one-dimensional Hamiltonian for the u coordinate: h = p 2 / 2 - ( 1/2p)e-flp2. The motion along u is that of a particle on a symmetric potential well at the "asymptotic" dissociation energy. This trajectory exactly spans the range --OD Iu Im. In the high-temperature limit, one can expand the exponent in the effective Hamiltonian H, to find that the dividing surface is a trajectory on the potential energy surface E a t the energy 1/2& Finally, the exact expression for the transmission coefficient (eq 2.19) may be rewritten in terms of the action along the optimal trajectory. Denote the configuration space length along the trajectory by s such that ds2 = d$ + du2. Then, eq 2.19 can be rewritten as
P = - S (27p
d ~ [ 2 ( t- Vb)]'/2
(2.32)
Canonical VTST is thus reduced to running classical trajectories of an effective 2 degrees of freedom Hamiltonian and looking for that orbit whose classical action is minimal with the additional constraint that the configuration space path of the orbit is a "good" dividing surface. The present optimization procedure is not restricted to dissipative systems. The variational solution for the canonical rate associated with any 2 degrees of freedom Hamiltonian may be found in the same manner. The canonical optimal dividing surface is a classical trajectory of a temperature-dependent Hamiltonian. In the present paper however, we will restrict ourselves to the more approximate but simpler and analytic solution obtained by optimizing only the exponent appearing in eq 2.19 and applying it to the cubic oscillator. This is presented in the next section. 3. Cubic Oscillator a. Theory. The cubic oscillator is an especially interesting application for VTST since nontrivial bounds are found only when using a curved dividing surface. Inspection of eq 2.19 shows that a linear choice for the dividing surface will usually lead to a divergent result because of the nonpositive character of the anharmonicity. On the other hand, the cubic form is still simple enough to enable a clear demonstration of the variational method. As a first step we define some reduced variables. The length scale of the problem is the cubic potential parameter q,, so henceforth, in this paper, all coordinates will be scaled accordingly; i.e., g = g / q o and u = u/q,. It is also useful to define a reduced friction parameter x (3.1) and a reduced barrier height The variational solution for the dividing surface, as obtained from eqs 1.3 and 2.21 leads to a quadratic equation uoog =
-3/x(uOog + u1aI2
(3.3)
from which it is clear that the dividing surface g(u) is negative for all u. Using the explicit notation uoog 5 -r2
(3.4)
Pollak we find that the dependence of r on u is given by the solution of a quadratic equation:
(
r = $)'l2[-l
+ (1 + ~ U , X U ) ~ / ~ ]
(3.5)
This solution has some interesting properties. For small u we find that r is linear in u, implying that the dividing surface g(u) is quadratic in u. For large, positive u we find that u,g uIu = 0; that is, the dividing surface coincides with the surface defined by q = 0. As stressed in the previous section, using q = 0 as the dividing surface everywheres leads to the TST rate as defined by Kramers; that is, r = (w0/2ir)eQ'. Here we see that in the critical range where the generalized bath coordinate u becomes large and would lead to a divergent (trivial) bound, the variational form automatically transforms to the very simple dividing surface q = 0, which is guaranteed to lead to a finite contribution. Finally, the variational solution given by eq 3.5 is well-defined only for u l u 1 -1/6x so that more work is needed before the dividing surface is well-defined, that is defined for all values of u. The problem associated with extension of the dividing surface to negative values of u is directly related to the value of the second variation of the exponent, as given in eq 2.22. As noted in the previous section, the solution of the variational equation (2.21) is valid only as long as the second variation is negative. For the cubic oscillator, this translates itself immediately to the condition (1 + ~ U , X U ) 1 ~ /0. ~ This means that for more negative values of the collective bath coordinate u there is no steepest descent variational solution to the variational problem defined by maximizing the exponent E [ g ] ;cf. eq 2.20. In practice, we find that a simple and useful choice is to use the variational result (eqs 3.4 and 3.5) for the dividing surface only for 0 I u Im (region I), while for --m I u I 0 (region 11) we choose the dividing surface g = 0. The integration over u is thus divided into two parts. The contribution of region I to the transmission coefficient P, denoted by PI, is obtained by insertion of the expression for the dividing surface in this region into the variational integral, eq 2.19. Defining the variable z = (6x)'i2r, one finds
+
r, um2(
1 + 22)] 1/2,-(Q'/
l6x3(x-I)) [3rP+z'(8x+4)+l 2xr2]
(3.6)
For region I1 one finds
(3.7) and the full transmission coefficient is the sum of these two terms: P = P, PI1 (3.8)
+
Note that at this point there are no divergences. Moreover, a steepest descent estimate of both integrals yields the minimum of the exponent at z = 0 with the result P = 1 and one has regained the Grote-Hynes expression for the rate. It is also evident that the contribution from region I1 is always smaller than expected from the Grote-Hynes theory, since in this region the nonlinear cubic term is positive definite. The major deviations from Grote-Hynes theory will come, if at all, from region I where the curvature is an important factor. b. Application with Ohmic Friction. The easiest case to analyze is when the friction is ohmic y ( t ) = 2yW) (3.9) where y is the friction coefficient and is also identical to the Laplace transform of the ohmic friction. In the following we will use the dimenskdess variable a E y/2w*. As shown by Kramers,' when the friction is ohmic, the reactive frequency is
_ A' - (1 f w*
,*)'P
-a
(3.10)
The Journal of Physical Chemistry, Vol. 95, No. 25, 1991 10239
Dissipative Cubic Oscillator I
-4
I
I
I
I
t
1 -6
I
-4
-2
2
0
4
6
uwp
TABLE I: Transmission Coefficients for the Cubic Oscillator Q' PI PI I P (A'/W')P (a) x = 1.1, a = 0.100504,h'/w' = 0.904534 1 0.90463 0.503473 0.496634 1.00011 3 0.502043 0.49804 1 .OO008 0.904609 5 0.501576 0.498478 1.00005 0.904583 7 0.501323 0.49871 2 1 .OO003 0.904566 9 0.501 158 0.498863 1.00002 0.904553 11 0.501039 0.498971 1.00001 0.904543 13 0.500948 0.499053 1 0.904535 15 0.500876 0.499118 0.999994 0.904528 17 0.500816 0.499171 0.999987 0.904522 19 0.500766 0.499216 0.999982 0.904517 21 0.500722 0.499254 0.999976 0.904513 23 0.500685 0.499287 0.999972 0.904509 25 0.500652 0.499316 0.999968 0.904505 27 0.500622 0.499342 0.999964 0.904501 29 0.500595 0.499365 0.99996 0.904498
(b) x = 1.75, a = 1.113 389,X'/w' = 0.377964
-6
I
-4
I
I
I
-2
0
I I
2
4
6
I
I
I
"WO I
-6
f
-6
I
-4
1
I
I
I
-2
I
2
0
I
4
I
6
%.9 Figure 1. Effective potential energy and dividing surface for the cubic oscillator. Panel a: x = 1.1, contours shown for the reduced energies -20,-15, -10,-5,0.002,5, 10,15, and 20. Panel b: x = 1.75,contours at the reduced energies -20,-15, -10, -5, 0.0025,1, 2.5,5, 10,15, and 20. Panel c: x = 2, contours the same as in panel b.
and the transformation matrix element uoo2is found (cf eq 2.6) to be (3.11) Note that in the strong damping limit a >> 1 this matrix element is inversely proportional to the damping squared. The reduced parameter x (cf. eq 3.1) is therefore bounded between the values 1 and 2. x=l+
(1
+
a a2)1/2
(3.12)
For the cubic oscillator, the effective potential appearing in the reduced Hamiltonian H* may be written in the dimensionless form: 21 x - l
x
,--w
Contour plots of this effective potential are shown in Figure 1 for
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 11 13 I5 17 19 21 23 25 27 29
0.65334 0.597536 0.576691 0.564786 0.556821 0.551014 0.546541 0.542962 0.540018 0.537542 0.535424 0.533587 0.531974 0.530545 0.529267
0.448387 0.466645 0.473 127 0.476782 0.479221 0.481003 0.482381 0.483489 0.484405 0.48518 0.485848 0.48643 0.486944 0.487403 0.487815
1.10173 1.06418 1.04982 1.04157 1.03604 1.03202 1.02892 1.02645 1.02442 1.02272 1.02127 1.02002 1.01892 1.01795 1.01708
0.416414 0.402223 0.396794 0.393676 0.391 587 0.390066 0.388896 0.387962 0.387195 0.386553 0.386004 0.38553 0.385 115 0.384748 0.384421
(c) x = 1.999,a = 22.3439,X * / w * = 0.022366 3 7.48542 0.429586 7.91501 0.177029 5.69027 0.452886 6.14316 0.1374 4.92848 0.461 51 5.38999 0.120554 4.4586 0.466468 4.92507 0.110155 4.12569 0.469819 4.59551 0.102784 3.87147 0.472289 4.34376 0.0971537 3.66792 0.474211 4.14213 0.0926441 3.49951 0.475766 3.97527 0.088912 3.35675 0.477058 3.833 81 0.085748 3.23348 0.478156 3.71163 0.0830154 3.12546 0.479 103 3.60456 0.0806206 3.02966 0.479933 3.50959 0.0784965 2.94386 0.480668 3.42452 0.0765938 2.86636 0.481324 3.34768 0.0748751 2.79586 0.481916 3.27777 0.0733116
the values of x = 1.1, 1.75, and 2, respectively. In addition, the variational dividing surface is superimposed on the plot. These three cases correspond to weak (a> 1) damping, respectively. In the weak-damping limit, the curvature of the surface is actually not too important; at p = 0 one must go to very large values of Q before reaching negative values of the potential. This is not the case for intermediate and strong damping. Note that the variational dividing surface approaches the line q = 0 a t large positive values of U. Some representative numerical results for the transmission coefficient are provided in Table I for thevalues of x = 1.1, 1.75, and 1.999. In the weak- and intermediate-damping range, one finds that the present VTST is practically identical to the Grote-Hynes result. As expected, PI is always slightly greater than while PIIis slightly smaller than but the sum of the two is at most only very slightly larger or smaller than unity. In the strong-damping limit, one finds large differences; the VTST gives a transmission coefficient which is substantially larger than 1. From the expression for the transmission coefficient in region I (eq 3.6) one notes that in the strong-damping limit for which uoo2> 1. As noted earlier, the variational surface derived from eq 2.21 is valid only as long as g'
