RESEARCH ARTICLE | SEPTEMBER 15 1977

A new tunneling path for reactions such as H+H2→H2+H 

R. A. Marcus; Michael E. Coltrin

J. Chem. Phys. 67, 2609–2613 (1977)

https://doi.org/10.1063/1.435172

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09 July 2024 12:16:11

A new tunneling path for reactions such as
H + H2-+H2 + Ha)
R. A. Marcus and Michael E. Coltrin
Department of Chemistry. University of Illinois. Urbana. Illinois 61801
(Received 20 May 1977)

The standard tunneling path in transition state theory for reactions such as H + H2-+H2 + H has been the
so-called reaction path. namely the path of steepest ascent to the saddle point. This path is now known to
give numerical results for the reaction probability which are in disagreement with the exact quantum
mechanical ones by an order of magnitude at low tunneling energies. A new tunneling path corresponding
to a line of vibrational endpoints is proposed. It is much shorter and is shown to give results in agreement
with the quantum ones to within about a factor of two. A semiclassical basis for choosing this new path is
given.

I. INTRODUCTION ly, the system will stop its motion in the s-direction at
the s =s* for which the energy barrier equals the total
Recent comparisons of quantum mechanical and tran-
energy, i. e., for which
sition state theory calculations for the colinear and
three-dimensional reaction rates of H + Hz - Hz + H have V1 (s*)+E o(s*)=E, (2.1)
revealed significant discrepancies between the two meth-
for a system in its lowest vibrational state.
ods. 1 These discrepancies occur particularly at low en-
ergies, where tunneling is very important. The quantum The coordinate perpendicular to the s-curve, a vibra-
mechanical rate is frequently of the order of ten to a hund- tional coordinate, is denoted by p, with p positive when
red times larger than the rate predicted by transition this coordinate is stretched, and negative when com-
state theory. Various numerical complex-valued classi- pressed. p equals a (signed) measure of the distance
cal trajectory studies have been made in this tunneling perpendicular to a tangent to the s- curve. The potential
region, and used in semiclassical calculations Z,3 and energy associated with p-motion at any s is designated

09 July 2024 12:16:11

more recently in a periodic-trajectory-transition-state V2 (p, s), where V2 (O, s) =0 on the s-curve. The maximum
(PTTS) formalism. 4 The results obtained using the vibrational amplitude at a given s is the p =Pmu which
PTTS show markedly improved agreement with the quan- satisfies
tum mechanical results. 4 (The semiclassical ones did
(2.2)
also, but they are not of the transition state theory type.)
The question arises whether there is some simple The family of points [Pmax(s), s1 satisfying (2.2) and for
physically intuitive modification of the usual tunneling which P is positive describes a curve, which we shall
call the t- curve.
calculation in transition state theory which yields good
agreement, without requiring the computation of actual The contour lines of a typical potential energy surface
classical trajectories. Such a method is described in for the H + Hz reaction are depicted us ing the usual
the present paper. It involves a new tunneling path for skewed axes 5 in Fig. 1. The. reaction path (the s- curve)
this H + Hz - Hz + H reaction, a path corresponding to the is shown as a solid line, and the t- curve as the dotted
vibrational limit during the motion. A semiclassical line. The central idea of the present paper is that a pre-
basis for the method is given. ferred way of tunneling is not along the reaction path,
the s-curve, but rather along a shorter path, the t-
II. TUNNELING PATH AND RESULTS curve described above. The tunneling along the t-curve
In tranSition state theory 5 it is customary to calculate starts from a point P for which s =s* and P =Pm ax
the "reaction path, " the curve of steepest descent pass- and continues along the t-curve to a corresponding point
ing through the saddle-point, and employ it as the tunnel- P' in the exit channel, i. e., the point for which (2.1) and
ing path. We let the coordinates along that path be de- (2.2) are again satisfied but in the exit channel in Fig.
noted by s, and the potential energy along this "s- curve" 1. The starting and end points on the t- curve, P and pI,
by V1 (s). For a colinear reaction in the tunneling region depend on the energy E, as in (2.1), and are given in
the vibrational energy of the lowest vibrational state in Fig. 1 for a particular E. If V denotes the potential en-
the transition state at s = sl is the zero-point energy, ergy along the t- curve then
Eo(sl). The translational energy available for tunneling V= V1 (s)+E o(s) (on t-curve) , (2.3)
at any s is then often chosen to be the total energy E min-
us V1 (s)+E o(sl). Actually, the effective potential energy as compared with V1 (s) on the s-curve.
barrier is V1 (s)+E o(s), where Eo(s) is the system's lo- If dq denotes an element of length along the tunneling
cal zero-point energy at a given s, and this barrier is path, then the imaginary part of the complex-valued
now frequently used instead of V1 (s)+ Eo(sl). Classical- phase integral which appears in a tunneling calculation is
J(E).

algupported by a grant from the National Science Foundation. J(E) = 1m f

P
P
' pdq/Ii (2.4)

The Journal of Chemical Physics, Vol. 67, No.6, 15 September 1977 Copyright © 1977 American Institute of Physics 2609
 2610 R. A. Marcus and M. E. Coltrin: New tunneling path

J.5

].0

2.5

2.0

l.5

l.0

WALL-Po.RTER SURFACE
Fo.R H + Hz - H2 + H
D.5
l.5 2.0 2.5 3.0 3.5 4.0 4.5
00. 0.10. 020
X IR. U. J
Etrons(eV)
FIG. 1. Plot of potential energy contours for the H + H2 - H2
+ H reaction using the Porter-Karplus surface. Solid line is FIG. 3. Same legend as Fig. 2, but for an SSMK Wall-Porter
potential energy surface. Exact results are from Ref. 10(b)
line of steepest ascent (reaction path). Dotted line is the t-
curve (limit of vibrational amplitudes in the given vibrational and the PTTS curve is obtained from Fig. 5 of Ref. 4(b).
state, here the zero-point state). The points P and p' denote
the initial and final tunneling points on the t-curve for a par-
ticular total energy. The corresponding tunneling points if Fig. 1. The J.l and scaling factors used in the present
tunneling occurred along the reaction path are Q and Q' • paper are the standard ones. 5,6
The semiclassical transmission coefficient (ratio of
outgoing flux of products to incident flux of reactants) is

09 July 2024 12:16:11

where 1m denotes "imaginary part of"; P is the compo- K(E). Using the results obtained by mapping the effec-
nent of the momentum along the path, e. g., on the t- tive potential onto one of parabOlic form, K(E) is given
curve it is by7
(2.5) K(E) =exp(- 2J)/(1 +exp(- 2J)] . (2.6)
where V is given by (2. 3). Il is a reduced mass whose The initial translational energy is E- Eo(- 00), where
value depends on the distance scaling factors used in Eo(- 00) is the initial zero point energy. A plot of K(E)
versus this translational energy, for the Porter-Karplus
surfaces and for the new tunneling path, is shown in Fig.
2, together with the quantum mechanical 9 and convention-
al transition state results. The results are seen to
agree quite well with the quantum ones, 9 and show con-
siderable improvement over the use of the conventional
path. In Fig. 3 the corresponding results for the SSMK
Wall-Porter surface10 ,l1 are shown. The results agree
with the quantum results10(b) to about the same accuracy
as before. An interesting approach to transition state
theory, given by Miller 4 et al., utilizes numerically
computed classical trajectories (periodic-trajectory-
transition-state method, PTTS). The results are given
in Figs. 2 and 3.

Po.RTER-KARPLUS SURFACE In the H + H2 - H2 + H reaction at the total energies giv-

Fo.R H + Hz ~ Hz + H en in Figs, 2 and 3 no excited vibrational states of H2 can
exist. In other systems for which excited vibrational
0.10 0.20.
states can exist in this tunneling region, one can compute
Etrons (eV)
a transmission coefficient Kn(E) for the nth Vibrational
FIG. 2. Plot of reaction probability vs initial translational state at the given total energy, using (2.6) with Eo(s) in
energy in the center of mass system for the H + H2 - H2 + H (2.1)-(2.3) replaced by En(s). The sum of the trans-
reaction, for the Porter-Karplus potential energy surface.
Curves are given for the exact quantum mechanical result
mission coefficients at the total energy E is then the sum
{Ref. 9), the usual transition state theory result, the present Ln Kn(E) over all energetically accessible initial states n
transition state theory result and a result (PTTS) which intro- in a microcanonical ensemble of initial states. Micro-
duces numerically computed periodic trajectories into a transi- canonical tranSition state theory, and the manner in
tion state theory [curve from Fig. 6 of Ref. 4(b»). which each Kn contributes to the reactive flux is de-

J. Chern. Phys., Vol. 67, No.6, 15 September 1977

 R. A. Marcus and M. E. Coltrin: New tunneling path 2611

scribed, in both the adiabatic and general forms of mi- during the trajectory. 14 The assumption presumes a high
crocanonical transition state theory in Ref. 12. enough vibrational frequency of the motion transverse to
the reaction path. With the assumption of vibrational
A theoretical basis for the present choice for the tun- ad iabati city, the value of the classically attainable s be-
neling path is obtained from the semiclassical arguments comes independent of the initial vibrational phase woo
given in the next section. This s* is then given by Eq. (2.1) in the case that the
vibrational state is the lowest one (and by the same equa-
III. SEMICLASSICAL ARGUMENTS tion with Eo(s*) replaced by En(s*), when the vibrational
In semiclassical theory the action variables are the state is any given state n). These trajectories thereby
.
classIcal analog of the quantum num b ers. 13(a)-13(f) 0 ne each reach the same vibrational turning point P at s*,
can obtain a wavefunction13 (c).13(d) for a collision system and then tunnel from there since it is the closest point
beginning in a given specified initial quantum state n, by to the classically allowed region of the products' chan-
using a set of classical trajectories having the desired nel. Thereby, for each WO one has the same value of the
initial action variables but uniformly distributed in initial complex-valued quantity f pdq in (3.2) from point P to
phase woo When this wavefunction is introduced into a point pi for each member of this family of trajectories,
well-known quantum mechanical expression for the S- a family whose members differ only in woo
matrix elements s",n one obtains an integral expression InEq. (3.2) one can now set n =n, since in the vibration-
for these elements. 13(d).13(g) The reaction probability
ally-adiabatic approximation all trajectories will have
(transmission coefficient) of a system in state n is the same value of n,
namely the initial value n. (When
the coordinates dWO denotes dwf, dwg, •• " nand n de-
(3.1)
note nl> 112, .,. and nl> n2, ••• ). The integral over p
can be integrated by parts, yielding f pdq - ~R' + p~RI,
where the sum is over all final states m of products.
PR being the translational momentum, since the vibra-
The S-matrix element can be written as (3.2) for a two
tional momentum vanishes at the end points of the above
coordinate system (the dwo becoming dwfdwg • •• , and the
integration path. p~ and p1 are the final and initial
preexponential factor becoming a determinant, for a
translational momenta. The trajectories beginning with
higher dimensional system):13(d)
different WO will all have the same value for this integral

2: )UiO=O
r Iaw/awOl
1 1
/2 exp [ - i J'I qdp because of the absence of a relative distortion of the tra-

09 July 2024 12:16:11

Smn = jectories in a vibrationally-adiabatic approximation. It
paths pi
can then be placed outside the integral over woo Because
of this lack of distortion the aw/awo in (3.2) can be set
+2lTi(;i- m)w- NlTi] dwO , (3.2)
equal to unity. If the imaginary part of f Pdq/n, namely
the value along the path between P and pI, is denoted by
w
where is a final angle variable (a final phase), a con-
J(E), Eq. (3.2) gives (3.3) for I s",n 12, after integration
stant for any trajectory, f qdp denotes a path integral over wO,
with q being a distance along the path if the coordinates
are Cartesian, and p being the local Cartesian momen- I smnl 2 = exp(- 2J). (3.3)
tum component along the path. (If other coordinates are Equation (3.3) presumes only a single traverse be-
used qdp denotes a sum Lk qkdPk over all coordinates k.) tween points P and pi, whereas one should really sum in
The integration limits are from the value (pi) of the mo- the right hand side of (3.2) over all traverses, as indi-
mentum p at a vibrational endpoint at some large initial cated by 4ath. (suitably renormalized to conserve flux,
separation distance Ri to that (PI) at a vibrational end- when there is a branching of the paths). For example,
point at some final separation R'. The integration path the system may go from P to pi, and return to the re-
is chosen to consist of three parts: first at a fixed actants' channel, or go from p to pi, return to P, re-
R(=Ri) from the vibrational end-point to some desired turn to pI, and then go into the products channel, and so
initial vibrational phase wO, then along an actual classi- on. There are an infinite number of such paths. In ef-
cal trajectory to the final specified separation distance fect, a sum over all these paths is obtained by mapping
R' (in the present case in the products' chaQnel) and fi- the tunneling problem between P and pi onto the para-
nal phase W, and then at fixed R' to a vibrational end- bolic barrier problem, and solving that problem, with
point at the R'.The first and third legs of the integra- the result that the tUnneling factor is given 7 by (3,4) in-
tion are performed in regions where the internal motion stead of (3. 3),
is separable from the translational motion, and so an in-
tegration path at fixed R can be chosen in those regions. I smnl 2 = exp(- 2J)/[1 + exp(- 2J)] (3.4)
N is related to the number of times the trajectory touch- Eq. (3.4) reduces to (3,3) when J is large, that is when
es one of the vibrational caustics (the one not joining the all paths but the single traverse path become unimpor-
initial and final vibrational end pOints). 12(d) The classi- tant.
cal mechanical analog of the final quantum number of the
To summarize, we have introduced into (3.2) the ap-
trajectory in the products' channel is n,
and the transi-
proximations of vibrational adiabaticity and a semi-
tion of interest for Smn is for n- m.
classical tunneling expreSSion (3,4), with the implication
The principal approximation which will be introduced of tunneling along the shortest path, namely between P
into (3.2) is one of vibrational adiabaticity, namely the and p'. The tunneling approximation (3.4) should intro-
assumption that the quantum number n remains constant duce very little error, since it has been numerically

J. Chern. Phys., Vol. 67, No.6, 15 September 1977

2612 R. A. Marcus and M. E. Coltrin: New tunneling path

tested. The third and final approximation which remains (M. E. C. ) Was the recipient of a University of Illinois
to be introduced is the choice of the optimum tunneling Fellowship.
path between P and pi for calculating J. The "best" path
is a dynamical one, namely the one which, by Hamilton's APPENDIX A. ACTION CALCULATED ALONG
principle of least action, has the least value of f Pdq be- ALTERNATIVE TUNNELING PATHS
tween the two points. 15(b) We have selected the t-curve,
the t- curve being one which involves tunneling in the s-
We consider the phase integral I:'
pdq along several
paths to compare with the value along the t-curve. The
direction and not in the p-direction. We have indeed ex-
principle of stationary action for fixed P, pI, and E is
amined a number of other paths and found the 1m! pdq
for those paths for the present reaction either to have
nearly the same or a larger value. Examples are given fJ t
p'
pdq =0 , (A1)
in Appendix A.
which implies that the variations (from the value along
A principal assumption, as already noted, is the vi- the best dynamical path) of its real part and of its imag-
brational adiabatic one. Actually, any reaction, even inary part are zero. The latter part determines the tun-
H + Hz - H2 + H, is at least somewhat vibrationally- non- neling probability [cf. Eq. (3.3)], and we focus attention
adiabatic. 14 For example, classical trajectories for this on it. Strictly speaking, the p in (A1) should be directed
reaction reveal that s* depends somewhat on -UP.16 Thus, along that path. We first conSider some paths for which
as a result of passing through the pretransition state re- this is not the case but which satisfy vibrational adia-
gion there has been some change in the vibrational action baticity.
variable of the p-motion before reaching s*, whereas
One family of curves is the following: (1) a path at
that action variable would be constant in a vibrational-
constant s* from p =Pmu. to P =kPmu where k is a constant
adiabatic approximation. Thereby, for some wO's the
less than unity, (2) a path with p(s) =kPmu(s) from that
s* is larger and for others smaller than that determined
s* to the s* in the exit channel, and (3) a path at that s*
by Eq. (2. 1). We have termed this vibrational nonadia-
from P = kP mu to p'. Only step (2) contributes to 1m! pdq,
baticity elsewhere the "nonadiabatic tail, ,,17 because
when P lies between its minimum and maximum classi-
some systems will pass over the barrier at energies
cally allowed values. At any point in step (2) the rele-
where in the vibrationally-adiabatic approximation they
vant value of p, P., in the integrand (the component along
could not. The agreement in Figs. 2 and 3 is neverthe-
the path) is {2J.,L[E- Eo(s)- V1 (S)]}I/2, since

09 July 2024 12:16:11

less seen to be quite reasonable.
(A2)

IV. DISCUSSION Thus, at any s in step (2) the p in the integrand, namely
P., is the same as the p on the t-curve, given by Eqs.
The new tunneling path is a simple path which provides
(2.3) and (2.5). However, the path along step (2) is
a considerably improved agreement with the quantum re-
longer than that along the t-curve, and thus the value of
sults, as compared with the standard tunneling path. We
1m! pdq is greater than that along the t-curve. For ex-
have neglected vibrational-nonadiabaticity in the pre-
ample, for the SSMK Wall-Porter surface at E= 0.3985
transition state region, as indeed do all quantum transi-
eV, J(E) along the s-curve is 5.81, whereas that along
tion state theories. Vibrational nonadiabaticity allows
the t-curve is only 3.65. (J is 1m! pdq/fi.)
some systems to pass sl with a vibrational energy less
than Eo(sl), and causes any transition state theory re- Another set of paths is that for which p(s)~Pm""(s).
sults at energies near E equal to V(s1) + Eo(s1) to be too Once again we first choose a three-step path: (1) a path
low. The error is not more than a factor of about two, at the initial s* from P =Pmu. to P =kPmu, where k is a
judging from the results in that region (largely not given constant greater than unity, (2) a path with p(s) = kPmu.(s),
in Figs. 2-3, but calculated). 18 from the initial s* to the s* in the exit channel, and (3)
at the final s* from p =kP mu to P =Pm"". Now all three
We have considered above a class of reactions involv-
steps contribute to ImJ pdq. For k =1. 01 and 1. 05, the
ing three centers of comparable (in the present case
values of J were 3.73 and 4.07, respectively, for the
equal) masses. One system of particular interest is the
cited E, thus once again exceeding the value of J =3. 65
transfer of a light particle between two heavy ones.
for the t- curve.
Here, the acute angle in Fig. 1 is so much smaller that
the exit and entrance channels are almost parallel. Dy- These results are summarized in Table I. The path
namically this syst€m is quite different, and it is planned along the t- curve is the only internally consistent path in
to discuss tunneling for such a system elseWhere. Table I: It alone has a zero component of velocity nor-
mal to it.
Finally, we should note that tunneling along a path
other than the standard reaction path was first employed Many other paths can be suggested, and a complete in-
by Johnston and Rapp, 19 who considered straight line vestigation of them would be equivalent to solving Hamil-
paths. ton's equations in the Vicinity of the saddle-point.
Among the classes of paths are (A3), chOOSing s = 0 to
lie along the bisector of the acute angle in Fig. 1.
ACKNOWLEDGMENT
p(s)=[1=Fa(l-ls/s*I)]Pmu.(s) (O<a<l). (A3)
We are pleased to acknowledge the support of this re-
search by the National Science Foundation. One of us With the minus sign, one would have p(s)~Pmu(s), and

J. Chern. Phys., Vol. 67. No.6, 15 September 1977

 R. A. Marcus and M. E. Coltrin: New tunneling path 2613

TABLE 1. Summary of phase integrals +m2)(m2+mS)]1/2. Here, (11/2)-0 denotes the acute angle in
along different tunneling paths. Fig. 1. The coordinates x, y have the property that they di-
agonalize the kinetic energy, and both have a single mass 11,
Description of patha ImJ pdq/If in the center of mass system of coordinates, i. e., the kinetic
energy in this system is (11 /2)(i 2 +y2), with 11 = ml (m2 + ms)/
P =0 (s-curve) 5.81
(ml + m2 + ms)·
P =kPmax (k < 1) >3.65 7Compare with J. N. L. Connor, Molec. Phys. 15, 37 (1968),
Eqs. (14), (15), (18), and (19). The transmission coefficient
P = Pmax (t-curve) 3.65
is the ratio of the transmitted to incident flux, and is calcu-
p=1.01pmax 3.73 lated from the wave functions in Eqs. (14) and (15), and found
to give the present Eq. (2.6). The mapping equations are Eqs.
P =1.05 Pmax 4.07
(18) and (19).
SR. N. Porter and M. Karplus, J. Chern. Phys. 40, 1105
aThese paths refer to three-step paths,
(1964).
but only the middle step between P and
9G. C. Schatz and A. Kuppermann (private communication from
P' is described in this column. The
G. C. Schatz); cf. Fig. 6 of Ref. 4(b); J. w. Duff and D. G.
value in the second column is the value
Truhlar, Chern. Phys. Lett. 23, 327 (1973).
of ImJ pdq/If for the entire path between
10(a) D. G. Truhlar and A. Kuppermann, J. Chern. Phys. 52,
p and P'. All results are for the SSMK
3841 (1970); (b) ibid. 56, 2232 (1972).
Wall-Porter surface for E=0.3985 eV.
l1Reference 10 gives a fit to the results of a calculation of the
Hs surface in 1. Shavitt, R. M. stevens, F. L. Minn, and M.
Karplus, J. Chern. Phys. 48, 2700 (1968) to the functional
in the vibrationally-adiabatic approximation the Imp in formgiveninF. T. WallandR. N. Porter, J. Chern. Phys.
J(E) would still be the Ps given by (A2). The s-distance 36, 3256 (1962). The fit is actually to a scaled version of
part of the path length would be greater than that for the SSMK, scaled as per the suggestion of I. Shavitt, J. Chern.
Phys. 49, 4048 (1968).
t-curve and so J(E) would be larger. With the plus sign 12R. A. Marcus, J. Chern. Phys. 45, 2138 (1966). cf. Eqs.
of (A3), p(s)~Pmax(s). If one used zero velocity compo- (3) and (4).
nent normal to the path, thereby dropping the vibrational IsFor example, (a) M. Born, Mechanics of the Atom (Ungar,
adiabaticity (other than for the t-curve, for which a = 0), New York, 1960); (b) J. B. Keller, Ann. Phys. (N. Y.) 4,
p would be [2/l(E- V)]1/2, where Vis the potential ener- 180 (1958); (c) R. A. Marcus, Chern. Phys. Lett. 7, 525
gy on the path, and J(E) could readily be calculated. (1970); (d) R. A. Marcus, J. Chern. Phys. 59, 5125 (1973);
(e) W. H. Miller, Adv. Chern. Phys. 25, 69 (1974) and

09 July 2024 12:16:11

Calculations for these paths and for other systems will
references cited therein; (f) W. H. Miller, J. Chern. Phys.
be presented elsewhere.
54, 5386 (1971); (g) cf. integral expression in W. H. Miller,
We have not discussed energies where E> Vl(st)+Eo(st) J. Chern. Phys. 53, 3578 (1970).
but for which diffraction can occur when E is just above 14(a) J. O. Hirschfelder and E. Wigner, J. Chern. Phys. 7,
616 (1939), who considered a quantum system rather than a
this barrier. Here, the path which maximizes K is one
trajectory; (b) Nonadiabatic effects in the classical mechanics
for which the end pOints are imaginary rather than real, of reactions are discussed in R. A. Marcus, J. Chern. Phys.
and is not, therefore, between P and P'. The K(E) will 45, 4500 (1966). Vibrational adiabaticity has been used by
lie between the values of 0.5 and unity in this region. (c) M. A. Eliason and J. O. Hirschfelder, ibid. 30, 1426
(1959); (d) L. Hofacker, Z. Naturforsch. Teil A 18, 607
(1963); (e) R. A. Marcus, J. Chern. Phys. 43, 1598 (1966),
IE. g., D. G. Truhlar and A. Kuppermann, J. Am. Chern. which coined the term "vibrational adiabaticity, .. and (f) R.
Soc. 93, 184 (1970); D. G. Truhlar and A. Kuppermann, A. Marcus, J. Chern. Phys. 46, 959 (1967).
Chern. Phys. Lett. 9, 269 (1971): G. C. Schatz and A. 15Compare with H. C. Corben and P. Stehle, Classical Me-
Kuppermann, J. Chern. Phys. 65, 4668 (1976) and references chanics (Wiley, New York, 1964), 2nd ed., p. 170 Eq. (57.12)
cited therein; E. Mortensen, ibid. 48, 4029 (1968). 16Forexample, in Fig. 80fS. F. Wu and R. A. Marcus, J.
2(a) T. F. George and W. H. Miller, J. Chern. Phys. 56, 5722 Chern. Phys. 53, 4026 (1970), the trajectories for some ini-
(1972); (b) ibid. 57, 2458 (1972). tial phases are unreactive while others are reactive. The
SJ. R. Stine, Ph. D. Thesis, University of Illinois, 1974. second group thereby has no real-Valued s*. Unpublished
4(a) W. H. Miller, J. Chern. Phys. 62, 1899 (1975); (b) S. trajectory studies of J. R. Stine in this laboratory showed
Chapman, B. C. Garrett, and W. H. Miller, J. Chern. Phys. analogous effects. Compare with R. A. Marcus, Ber.
63, 2710 (1975). Bunsenges. Phys. Chern. 81, 190 (1977) for a qualitative
5S. Glasstone, K. J. Laidler, and H. Eyring, The Theory of discussion of nonadiabatic effects.
Rate Processes (McGraw-Hill, New York, 1941), p. 100. 17R. A. Marcus, Ref. 13(£) cf. Ref. 13(b).
6For a reaction of three atoms of masses m1> m2, and ms in IsJudging from the results near K(E) =0.5, these effects appear
a line, with rl denoting the distance between ml and m2 and to be larger for the Porter-Karplus surface than for the
r2 denoting the distance between m2 and ms, the coordinates Wall-Porter one.
x and y in Figs. 1-3 are defined byrl =x-ytanO, r2=cysecO, 1~. S. Johnston and D. Rapp, J. Am. Chern. Soc. 83, 1
with c = [ml(m2 +ms)/mS(ml +m2) ]1/2 and sinO = [mlmS/(ml (1961).

J. Chern. Phys., Vol. 67, No.6, 15 September 1977