3444
J. Phys. Chem. 1994,98, 3444-3449
Resonance State Approach to Quantum Transition State Theory Meishan Zhao and Stuart A. Rice’ Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 Received: May 21, 1993: In Final Form: July 23, 1993”
We show, within the framework of the reaction coordinate representation of the dynamics of a chemical reaction, that application of a complex scaling transformation to the reaction coordinate leads naturally to the identification of transition states as scattering resonances. Our analysis also leads to the definition of a resonance width operator, whose expectation values are the lifetimes of the scattering resonances. I. Introduction The transition-state theory (TST)l-lO of reaction rate is the most widely used description of the dynamics of chemical transformation, hence has been intensively studied for more than half a century. The original development of the theory made use of classical concepts to define both the multidimensional surface in phase space that separates reactants from products and the flux of phase points through that surface.’-’ It was realized some time ago that the transformation of that classical analysis of the rate of reaction into a quantum analysis of the rate of reaction requires the solution of a fundamental problem. In the succinct language of Truhlar and Garrett:” “In quantum mechanics, if we localize a system in a hypersurface, we have complete uncertainty about the momentum normal to that hypersurface (i.e., in the direction of the reaction coordinate) and hence about the flux through that hypersurface.” The many different developments of a quantum mechanical version of transitionstate theory each emphasize a particular view of the approximations which can be used to handle this difficulty and the complications associated with the multibody aspects of the dynamics. It is worth noting that, although the most important part of the spectrum of the molecular complex that is the dynamical intermediate in the TST description of the reaction rate is dominated by scattering resonances, most of the formulations of the theory do not make a direct association between scattering resonances and the transition state. Two exceptions to this avoidance of the direct description of the role of scattering resonances are the derivation of TST proposed by DalgarnoI2 and that proposed by Truhlar and Garrett.” Dalgarno uses the Breit-Wigner form for the many channel reactive cross section with the assumption that the interferenceterms between different levels (resonances) of the collision complex are either negligibly small or cancel on average because of the random signs and magnitudes of the matrix elements. In this formulation, transitionstate theory emerges as a method for calculating the mean scattering resonance width and the collision complex partition function. Truhlar and Garrett reformulate the transition-state theory in terms of compound state resonances associated with the transition-state region, starting from the notion that each threshold is associated with a resonance. The accuracy of this reformulation is supported by the results of several recent numerical studies which show, among other studies, that the energy structure of the cumulative reaction probability is governed by a discrete spectrum of broad featuresthat can be associated with scattering resonances, each identified with a transition ~tate.13-1~ In this note we adopt the viewpoint of Truhlar and Garrett and show that there is a straightforward way of developing the identification of transition states with resonances. In particular we show that, after adoption of the framework of the reaction 0
coordinate description of chemical dynamics, complex scaling of the reaction coordinate generates an extended Hilbert space in which the transition states are well-defined scattering resonances. Our analysis leads to the definition of a resonance width operator whose expectation values are the lifetimes of the transition states. 11. Complex Scaling Transformation of the Reaction
Coordinate The complexscaling transformation’*-21of a dynamical system is based on replacement of the ordinary coordinates with complex coordinates. The transformation is carried out by use of the unitary scaling operator
O(e) = exp{i30)
(2.1)
where 3 is the generator of the scaling transformation and 0 is a positive angle parameter. The operator 3 is self-adjoint. Application of U(e) to the system wave function scales the coordinates such that
-
r r exp(i6) (2.2) In thecollision complex which transition statetheory is designed to describe all atomic motion except that along the reaction coordinate is bounded. We choose to apply complex scaling only to the reaction coordinate, denoted s. Use of (2.1) then leads to
O(e) +(s,x)
= eie/2+(seie,x)
(2.3)
where xrepresents all coordinates except the reaction coordinate. The specific form of the dilatation operator which generates a transformation of only the reaction coordinate can be obtained from (2.1)-(2.3). It can be shown that d
-[&e) dB
+(s,x)Ie=o = i3+(s,x)
(2.4)
Then, using (2.3), we obtain
with s’ = s exp(it9). For t9 = 0 we have
Abstract published in Advance ACS Abstracts, September 15, 1993.
0022-3654/94/2098-3444%04.50/00 1994 American Chemical Society
‘2[Adss + s-$]+(s,x)
(2.6)
Quantum Transition-State Theory
The Journal of Physical Chemistry, Vol. 98, No. 13, I994 3445
Comparison of eqs 2.4 and 2.6 yields
expression displayed in eq 2.10, we find
+ sz]
3 = -[-s 1
d d 2 ds If one defines the conjugate momentum operator for the reaction coordinate s as
Let
e a =--rag + Sj,], 2h
h d I%=?&
eq 2.8 can be written in the form
= &UP + SPSI
(2.9)
We now substitute (2.9) into (2.1) to obtain an explicit expression for the complex scaling operator:
O(e) = exp{i$e) = e x p-el z p g + sfis])
ps = h k ,
h t
h
(3.9)
T>e copplex scaling transformation requires that the operators A and B have the same 1evl: of priority in acting on the original wave function. Because A and B do not commute, we apply the identity exp(A) exp{b) = e x p p
+ B + i[a,b])
(3.10)
(2.10)
It is worth noting that (2.8) assumes that s can be treated as a locally Cartesian coordinate and that then (2.9) is just the symmetrizedquantum mechanical relation between the classical variables:
S = (ikg),
B =-
(2.11)
If we substitute the locally Cartesian coordinate approximation to the reaction path Hamiltonian operator k(s,x) = (1/2p)(f12 +$:) into (3.10), the commutator of Aand
+ V(S,X)
(3.11)
B assumes the form
When s is not a locally Cartesian coordinate, (2.9) remains valid but path curvature corrections must be included in the definition of Ps.
III. Resonance Width Operator A scattering resonance is conveniently described as a state with finite lifetime; for the cases of interest to us it can be represented as a superposition of bound and continuum states with complex energy?
E;= = E, - iF,/2
(3.1) where E, is the energy of the nth isolated narrow resonance and r, is the full width at half-maximum (fwhm) of this resonance. Thecomplexscaling transformationdescribed in the last section localizes the resonance states and isolates them from the continuum part of the spectrum. Indeed, the transformed scattering resonance acts like a bound state with square integrable wave function. Under the complex scaling operation the Hamiltonian of the system assumes the form
k(e)= O ( e ) k P ( e )
(3.2)
with the associated time-independent characteristic equation
fW)lW) ) = EIW))
(3.3)
where E is a complex eigenenergy. The time evolution of the system after complex scaling is described by
a ih$He,t))
= ri(e)lW,t))
(3.4)
which has the solution
[*(e,?))
= exp(- $(e)t}p(e,t=o))
(3.5)
Applying the complex scaling transformation defined in (2.10) to thereactioncoordinates,the timeevolutionof the wave function (3.5) becomes18920 +(seie,x,t) = e-i8(e)r/h+(seie,x) = e4(~4r/2)r/n +(seie,x)
(3.6)
the left-hand side of which can also be written +(seie,x,t) = O(e)e-i8r/he1(e>t q e ) +(s,x)
(3.7) Now replacing the complex scaling operator in (3.7) with the
where p is the reduced mass of the system. By using the identities @:S ,I
[V(s,x),$~= ih- v(s'x) as
= -i2h$,,
(3.13)
Equation 3.12 reduces to
Finally, inserting (3.14) into (3.8) and comparing the result with the right-hand side of (3.7), we obtain an operator corresponding to the imaginary part of the energy, namely
f(e) = - +,B] h
(3.15)
which is the resonance width operator 2
f(e) = 2e[$
- l$WS,X)] as
(3.16)
Equation 3.16 is the central result of our analysis; it is valid in the approximation that curvature of the reaction path has negligible dynamical effects. We note that the first term on the right-hand side of (3.16) is the normal kinetic energy operator conjugate to the reaction coordinates, and the second term is the virialof the systemdefinedwith respect to the reactioncoordinate. A brief discussion of the influence of curvature of the reaction path on the reaction path Hamiltonian will be found in section IV. IV. Quantum Mechanical Transition States and Resonances
We now examine some aspects of the behavior of a system described by the reaction path Hamiltonian (3.1 1). In general
3446 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
the reaction path is curved and the appropriate form of the Hamiltonian operator is23.24
where K ( S ) is the curvature of the minimum energy path and ),(s,x) is a function of s and x. Expanding the curvature term in a Taylor's series permits representation of the Hamiltonian operator in the form24 fi(s,x) = (1/2r)(pZ
+a:) + U(s,x)
(4.2)
Zhao and Rice the complex scaling transformation applied to the reaction coordinate does not alter the spectrum of the bounded motion. We now consider the continuum spectrum associated with motion along the reaction coordinate. We assume that there is a potential barrier along the reaction path. This barrier is, in general, a saddle point on the potential energy surface. As usual, we identify the (classical) transition state with the saddle point configuration. We can expand the potential energy in the vicinity of the top of the barrier using the saddle point as origin, whereupon the potential energy operator for the ground level of the transition state can be written in the form U(s,x) = V,'
where the effective potential U(s,x) is defined by
+ ' / , ~ k ~ , +' '/2k$ + Ul(s,x)
(4.10)
n
U(s,x) = V(s,x)
+ Vc(s,x)
(4.3)
If the so-called minimum energy path approximation is used K = 0 and V, = 0, which then leads to the reaction path Hamiltonian (3.1 1). In general the curvature K is not zero, and V, is also not zero. Furthermore, if we use the approximation
B,Z(S,O)
= - V,(~,S)I in (4.4), where E is the total energy and V, is the vibrational adiabatic potential curve, then eq 4.2 also reduces to (3.1 1). In the most general case, if the vibrational adiabatic approximation is not valid, the curvature correction to the kinetic energy operator forces us to use a more complicated analysis of the dynamics. For our present purposes we assume that it is not necessary to directly evaluate the consequences of reaction path curvature. As we have defined it, the complex scaling transformation is applied only to the reaction coordinate. By definition, when the reaction coordinate representation is used to describe a reacting system, all motions except along the reaction coordinate are bounded. Let the wave function for the system, in the reaction coordinate representation, be +bn(s,x),where n is a collection of quantum numbers for the bounded motions of the collision complex. Of course, for the bounded motions, Pandr are bounded. Then, by the virial theorem,25 the expectation value of the kinetic energy associated with the bounded motion is equal to its virial:
( T ) = '/,(r.VU)
(4.5)
Applying (4.2) to (3.16) yields
where VO' is the saddle point potential energy, k, < 0 and &(s,x) is the contribution to the potential energy from anharmonicity and the coupling of the reaction coordinate to the other degrees of freedom. Using (4.10) in (3.16) leads to
with K, = -k, > 0. Equation 4.1 1 is an exact expression for the resonance width operator of the transition state. It is obvious that theexpectationvalueof thisoperatordoes not vanish. Indeed, the zero-order expectation value of this resonancewidth operator at the transition state is
which is obtained by taking 0 = 1 rad. It is relatively straightforward to obtain a more accurate value for r by direct evaluation of (4.1 1). The zero-order approximation to 'I shown in (4.12) has an interesting interpretation. We note that the quantity whose expectation value appears on the right-hand side of (4.12) is the Hamiltonian of a harmonic oscillator. Thus, in the zero-order approximation, the complex scaling of the reaction coordinate converts the unstable motion at the top of a parbolic barrier along the reaction path into an imaginarystable harmonic motion. Far from the transition state the potential energy surface is flat, so that
?(e) = 2e#472r,
dV(s,x)/ds = 0
(4.13)
In this regime the imaginary part of the energy is, basically, the kinetic energy for motion along the reaction coordinate, and the continuum spectrum remains a continuum. V. A Simple Example
Consider, as a simple example, the linear harmonic oscillator
a2 + -ks2, 1 fi=- 2m 2
k
>0
(4.7)
To illustrate the calculation of the expectation value of the resonance width operator in a simple but nontrivial example, we consider passage of a particle over the modified Eckart potential barrier5914926 (see Figure 1):
where k is the force constant and m is the mass of the particle. Substitution of (4.7) into (3.16) yields the decay width operator:
In an eigenstate of harmonic motion the expectationvalues of the kinetic energy and the potential energy are equal, so
hence, as expected, the expectation value of the width operator (4.8) vanishes. In general, the expectation value of the width operator for a bound state motion vanishes or, put another way,
where a and bare parameters and VOis the barrier height. When a = 0, eq 5.1 reverts to the original Eckart potential barrier,26 which has often been used to model the potential barrier to a chemical reaction.5~26.27With suitable choices for the parameters (5.1) approximates the barrier in the collinear model of the reaction between H and H2. We note that when a is nonzero positive (5.1) describes a double barrier which is symmetric with respect to s = 0 (see Figure 1). Friedman and Truhlar14 have used (5.1) to study the relationship between chemical reaction thresholds and resonances; the resonance lifetimes were calculated using Smith's lifetime
Quantum Transition-State Theory V(s)
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3447
(meV)
(a.u.1
(a.u.)
Veff
-6
-4
-2
I
2
(5)
(meV)
Figure 1. (a) Modified Eckart barrier (5.1) when a = 0. (b) Modified Eckart barrier when CY = 0.68.
Figure 2. (a) Effective potential (5.4) when at a = 0. (b) Effective potential (5.4) when a = 0.68.
matrix theory.28 We now examine the use of (3.16) for the same purpose. To be able to compareour results with those of Friedman and Truhlar we choose the same potential parameters, namely, p = 1224.5me, VO= 0.08Ehand 0 = 1 . 0 ~ 0 The . parameter a is varied to change the shape of the potential barrier. To apply (3.16) to (5.1), we define an effective potential
expansion of (5.4) near these two local minima reads
so that
Veff(S)
= -Vb
1 1 + jka(lsl - a)' + jjC3(lSl -
+ ...
(5.5)
where Vb 1 0 is the height of the barrier between the local minima located at s = -a and s = a and
We consider first the harmonic approximation
(5.3) Substitution of ( 5 . 1 ) into ( 5 . 2 ) yields
Veff(s)
= -Vb
+ '/&,((SI
(5.7)
Now, when s = 3: f a we have Vedfa) = - Vb, and when s = 0 we have V,fr(O)= 0. The first condition is satisfied by (5.7), but the second condition is not unless Vb = '/2kbU2
which is an even function of s. Note that the potential energy is zero when s = 0 and that there are two local minima symmetrically located on each side of s = 0 (see Figure 2). An analytic calculation of the eigenvalues of f ( 8 ) with the effective potential displayed in (5.4) is difficult because Veri is a complicated function of s. However, we expect the ground state supported by Vem to make the dominant contribution to the resonance lifetime. In that case the use of a siFple approximation permits analysis of the expectation value of I'(8). In particular, when a = 0 the region of the effective potential important to defining the ground state energy is well approximated as a quadratic function (see Figure 1). When a # 0 the effective potential has the shape of a double well. The Taylor series
- 0)'
(5.8)
Clearly, (5.8) implies k, = k b . For our effective potential direct evaluation shows that, because of the anharmonicity correction near the minima, k b is close to but not equal to ka. Then a better approximation to (5.4) is obtained by use of the effective force constant
b'
'/Zkefd
(5.1 1)
We adopt (5.10) for our calculations;the values of the parameters used are listed in Table I.
Zhao and Rice
3448 The Journal of Physical Chemistry, Vol. 98, No. 13, 199'4 TABLE I: Effective Potential Parameters a 0.00 0.17 0.32 0.49 0.68
)v "b 0 0.00 3.07 13.0 26.1 41.7
a (ad o.Oo0 OOO 0.213 596 0.282 341 0.315 211 0.335 133
wen ( a 4 0.002 857 70 0.002 809 86 0.004 164 89 0.005 076 92 0.005 865 39
VI. Discussion
TABLE II: Resonance Lifetimes At (femtosecondsP ____~_______
0.00 0.17 0.32 0.49 0.68
ref 14
this work
33 43 58 91 180, lob
16.9 42.2
58.0 98.3 178,8
We note that Truhlar et al.11J4 defined the lifetime of a resonance state as At = h/lm(E), where E is the resonance energy. This definition isrelated tothatdisplaycdin(5.17) byAt= 27. bTherearetworesonancts in this system. These results are estimated from the data in Figure 2 of ref 14.
With (5.10), the eigenvalue equation for the width operator becomes
The solutions to this double oscillator equation are discussed in standard texts.29 It is found that the eigenfunctions of (5.12) are the parabolic cylinder functions and that the eigenvalues can be extracted from a system of coupled nonlinear equations.29 When (5.13) it can be shown that the fundamental frequency for the double oscillator system is w
(
= 20,ff *)'I2
exp(
T k f f
-5 ) h %ff
(5.14)
where the single oscillator frequency is, as usual, (5.15) In this limiting case the ground-state expectation value of the width operator is
from which we extract the resonance lifetime (5.17) We now consider the calculation of the resonance lifetimes for a range of values of a (see Table 11). For the case a = 0, u = 0,VO= 0, (5.12) reduces to the equation of motion of a harmonic oscillator with ken = k, = kb, which leads to w = wen. For the case a = 0.17, VOis very small. Using the potential parameters in Table I, we find that theconditionstated in (5.13) is not satisfied, hence the approximate double oscillator frequency (5.14) is not usable. By direct calculation we find, in thiscase, w. = 0.003 427 5 au and O b = 0.002 010 7 au, where w, is the frequency at s = a, and Wb is the frequency at s = 0. A simple approximation to the resonance lifetime is then 7 = 1/wa 1/wb. For all other values of a we have calculated the frequency from (5.14) and the resonance lifetime from (5.17). The results of our calculations are listed in Table I1 along with those calculated by Friedman and Truhlar using lifetime matrix theory.26 It is readily seen
+
that the resonance lifetimes calculated from our approximate treatment of the expectationvaluesof the resonance width operator and those calculated by Friedman and Truhlar are in excellent agreement, except for the case a = 0. We have not been able to track down, hence do not understand, the discrepancy of a factor of 2 in this case.
The usual formulation of quantum mechanics requires that operators whose expectation values represent observables be self adjoint, hence have real eigenvalues when standard boundary conditions are imposed on the solutions to the Schriidinger equation. In this formulation the energy spectrum of the system is always real, and it is not possible to understand or describe the properties of decaying states.30 The fact that states with finite lifetime can be described as scattering resonances, provided the boundary conditions are appropriately changed, was first pointed out by D i r a ~ . ~It' has come to be realized that the change in boundary conditions is just one of many ways of extending the Hilbert space so as to include the representation of states with finite lifetime. Since the analysisof the rate of a chemical reaction naturally leads to the discussion of states with finite lifetime, it is advantageous to use a formalism that permits representation of such states. The complex scaling transformation, sometimes called the dilation group analysis, is one of the modern methods of extending Hilbert space. We have shown in this brief comment that application of complex scaling analysis to the description of the rateof a chemical reaction avoids the fundamentaldifficulties alluded to by Truhlar and Garrett and yields a new result, namely a precise definition of the resonance width operator whose expectation values are the lifetimes of the transition states.
Acknowledgment. This work has been supported by a grant from the National Science Foundation. The authors thank the referees of this paper for several helpful comments. References and Notes (1) Wigner, E. P. Phys. Chem. (Munich) 1932,819, 203. (2) Eyring, H. J. J . Chem. Phys. 1935, 3, 107. Glasstone, S.;Laidler, K.;Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. Eyring, H.; Walter, J.; Kimball, G.E. Quantum Chemistry; John Wiley & Sons: New York, 1944. (3) Evans, M. G.;Polanyi, M. Trans. Faraday Soc. 1935, 31, 875. (4) Eliason, M. A.; Hirschfelder, J. 0. J . Chem. Phys. 1959, 30, 1426. (5) Johnson, H. S.Gas Phase Reaction Rate Theory;Ronald Prcss: New York, 1966. (6) Marcus, R. A. J. Chem. Phys. 1966,43,1598,2138. Marcus, R. A. J. Phys. Chem. 1979,83,204. (7) Truhlar, D. G. J . Chem. Phys. 1970, 53, 2041. Truhlar, D. G. J . Phys. Chem. 1979,83, 188. Garrett, B. C.; Truhlar, D. G.J. Phys. Chem. 1979,83,200,1079. Truhlar, D. G.; Isaamn, A. D.; Garrett, B. C. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Prcss: Boca Raton, FL, 1985. (8) Miller, W. H. J . Chem. Phys. 1974,61,1823; 1975,62,1899; 1975, 63.1 166. Miller, W. H. Faraday Discuss. Chem. Soc. 1977,62,40. Tromp, J. W.; Miller, W.H. J.Phys. Chem. 1986,90,3482. Miller, W. H.;Hemandez, R.; Handy, N. C.; Jayatilaka, D.; Willetts, A. Chem. Phys. Lett. 1990,72, 62. (9) Eu,B. C.; Ross,J. J . Chem. Phys. 1966,44, 2467. Coulson, C. A.; Levine, R. D. J. Chem. Phys. 1967,47, 1235. Pechukas, P. In Dynamics of Molecular Collisions, B.;Miller, W. H., Ed.; Plenum Press: New York, 1976. Kuppermann,A. J . Chem. Phys. 1979,83,171, Christov, S . 0.Collision Theory and Staristical Theory of Chemical Reactions; Springer-Verlag: Berlin, 1980. Pollak, E. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985. Bowman, M. J. Adu. Chem. Phys. 1985, 61, 115. (10) Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper & Row: New York, 1987; p 89. (11) Truhlar, D. G.;Garrett, B. C. J. Phys. Chem. 1992, 96, 6515. (12) Dalgamo, A,; Lewis, J. T. Proc. R . Soc. London Ser. A 1955,233, 70. Du, M. J.; Dalgarno, A.; J a m i a n , M. J. J. Chem. Phys. 1989,91,2980. (13) Atabck, 0.;Lefebvre, R.; Garcia Sucre, M.; Gomez-Llorente, J.; Taylor, H. S . Int. J . Quantum. Chem. 1991, 40, 211. (14) Friedman, R. S.;Truhlar, D. G. Chem. Phys. Lett. 1991, 183, 539. Chatfield, D. C.; Friedman, R. S.;Truhlar, D. G.;Garrett, B. C.; Schwcnke, D. W. J . Am. Chem. SOC.1991, 113,486.
Quantum Transition-State Theory (15) Schatz, G. C. J. Chem. Phys. 1989,90,3582,4847. Schatz, G. C. J. Chem. Soc., Faraday Tram. 1990,86, 1729. (16) Darakjan, Z.; Hayes, E. F.; Parker, G.A.; Butcher, E. A.; Kress, J. J. Chem. Phys. 1991, 95, 2516. (17) Chatfield, D. C.; Truhlar, D. G.; Schwenke, D. W. Faraday Discuss. Chem. Soc. 1991,91,289. Chatfield, D.C.; Friedman, R. S.;Schwenke, D. W.;Truhlar,D.G. J.Phys. Chem. 1992,96,2414. Chatfield,D. C.;Friedman, R. S.;Lynch, J.; Truhlar, D. G. Faraday Discuss. Chem. Soc. 1991,91,398. Chatfield, D. C.; Friedman, R. S.;Lynch, J.; Truhlar, D. G. J. Phys. Chem. 1992, 96, 57.
(18) Obcemea, Ch.; Brandas, E. Ann. Phys. 1983, 151, 383. (19) Reinhardt, W. P. In Mathematical Frontiers in Computational New York, 1988; Chemical Physics; Truhlar, D. G., Ed.; Springer-Verlag: . p 41. (20) Chatyidinmitriou-Dreismann, C. A. Ado. Chem. Phys. 1991,80,201, (21) Chu, S.In Resonances in Electron-Molecular Scatrering, van der Waals Complexes, and Reactive Chemical Dynamics; Truhlar, D. G., Ed.; American Chemical Society: Washington, DC, 1984.
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3449 (22) See for example: Lane, A. M.; Thomas, R. G.Rev. Mod.Phys. 1958, 30,257. Newton, R. G. Scattering Theory;McGraw-Hill: New York, 1966. Taylor, J. R. Scattering Theory; John Wiley & Sons: New York, 1972. (23) Marcus, R. A. J . Chem. Phys. 1 9 6 6 , 4 4 4 9 3 . (24) Garrett, B. C.;Truhlar, D. G. J.Phys. Chem. 1982,86,1136. Skodje, R.T.; Truhlar, D. G.;Garrett, B. C. J. Chem. Phys. 1982, 77, 5955. (25) Marion, J. B. Classical Dynamics of Particles andSystems, 2nd ed.; Academic Press: New York, 1970. (26) &kart. C. Phvs. Rcu. 1930. 35. 1303. (27j Scc for example: Truhlar, D. G.;i
