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Comparison of transition state theory with quantum scattering theory
for the reaction Li+HF+LiF+H
C.-Y. Yang and S. J. Klippenstein
Chemistry Department, Case Western Reserve University, Cleveland, Ohio 441067078
J. D. Kress and R. T Pack
Theoretical Division (T-12, MS B268), Los Alamos National Laboratory, Los Alamos, New Mexico 87545
G. A. Parker
Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019
A. Lagan&
Dipartimento di Chimica, Universitli di Perugia, Perugia, Italy
(Received 13 October 1993; accepted 21 December 1993)
The validity of transition state theory is examined for the bimolecular reaction of Li+HF-+LiF+H.
Accurate three-dimensional quantum scattering theory calculations of the cumulative reaction
probability are reported for energies ranging from threshold (0.255 eV) up to 0.600 eV and a total
angular momentum J of 0. Transition state theory estimates of the effect of both the entrance and
exit channels on the cumulative reaction probability are reported for the same energy range and J
value. The transition state theory results are found to provide an accurate description of the
smoothed energy dependence of the cumulative reaction probabilities with a maximum
disagreement between the two calculations of about 25% arising at the highest energy considered of
0.6 eV.
I. INTRODUCTION
Recent progress in quantum scattering theory has al-
lowed for the calculation of accurate three-dimensional cu-
mulative reaction probabilities for a variety of reactions.1-8
One of the important values of these cumulative reaction
probabilities is as a means for testing the validity of various
approximate theories of reactivity. Transition state theory
(TST) is one of the most often employed approximate theo-
ries of kinetics.‘-” Thus, there is considerable importance to
obtaining an accurate understanding of the limits of validity
of TST. Previous comparisons of TST with experiment have
generally suggested that TST does indeed provide accurate
estimates of rate constants. However, in these comparisons,
the generally limited accuracy of potential surface informa-
tion makes it difficult to obtain an absolute test of the TST
assumptions.
In this article, we present accurate three-dimensional
quantum scattering theory calculations of the cumulative re-
action probability (CRP) and the corresponding statistical es-
timates for the Li+HF+LiF+H reaction. The same potential
energy surface is employed in these two calculations, thereby
allowing for an absolute testing of the assumptions inherent
in the statistical calculations. This reaction presents an inter-
esting case for study since the exit barrier and entrance chan-
nel are nearly thermoneutral. This near equivalence in energy
results in an important effect on the reaction rate from both
the entrance and exit channels. Furthermore, there are impor-
tant tunneling effects for the motion in the neighborhood of
the exit channel barrier.
In Sec. II, a description is given of the potential energy
surface employed in these calculations. In Sec. III, the
coupled-channel hyperspherical coordinate based approach
employed in the quantum scattering theory calculations is
briefly described. The present statistical calculations employ
a variational treatment for the entrance channel and a tunnel-
ing and anharmonicity corrected quantized number of states
for the exit channel, as described in Sec. IV. The results for
the two different calculations are presented in Sec. V and
concluding remarks are made in Sec. VI.
II. POTENTIAL ENERGY SURFACE
The present calculations employed a modification’2 of
the potential energy surface described in Ref. 13. A sche-
matic diagram of the energetics along the reaction path is
provided in Fig. 1, while in Table I the structures and ener-
gies of the stationary points on this surface are provided. The
zero of energy is taken to be the reactants Li+HF in their
equilibrium configuration. The reaction is classically 0.157
eV endothermic and proceeds through a complex with a well
depth of 0.302 eV. The formation of the complex from reac-
tants is barrierless while its formation from products in-
volves a primary barrier of 0.182 eV relative to reactants. In
addition, there are a secondary well and barrier at larger
separations with energies of 0.113 and 0.176 eV relative to
reactants, respectively.
The inclusion of quantum zero-point energy changes this
picture dramatically as also indicated in Fig. 1. In particular,
the overall reaction becomes exothermic by 0.041 eV and the
primary barrier height is reduced to 0.048 eV relative to the
reactant’s zero-point energy. The small value for this zero-
point corrected barrier height suggests that bottlenecks in
both the formation of the complex from reactants and the
decay of the complex to products may be important. At low
energies, the exit channel barrier provides the dominant
bottleneck due to its slightly positive value. Furthermore, at
low energies the transition state for the entrance channel is at
J. Chem. Phys. 100 (7), 1 April 1994. 0021-9606/94/l 00(7)/4917/8/$6.00 0 1994 American Institute of Physics 4917
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4918 Yang et a/.: Comparison of TST and QST for Li+HF
+o.sos ov _
to..?5srv - +a917
&,A-o.~~~ et- _ +oJI* l v
+o.owev ~~~~167*v
/
i
Li+HF LgH / ;
LiF+H
t ,
Entrance
Channel
Primary Second
Barrier Barrier 22x4
FIG. 1. Schematic diagram of the minimum energy path for the reaction
Li+HF-+LiF+H. The bars correspond to the zero-point corrected energies
at the stationary points along the reaction path. The zero-point energy cor-
rection from the bending motion is neglected where the * appears due to the
low frequency hindered rotor nature of this mode at that location.
quite large separation distances and as a result the energy
levels for the bending mode are very closely spaced. Corre-
spondingly, again at low energies, the entropy for the en-
trance channel transition state is much greater than that for
the exit channel transition state where the bending energy
levels are more widely spaced. However, as the energy in-
creases, the entrance channel transition state moves in to
closer separation distances where the bending energy levels
are more widely spaced and the entropy gradually becomes
comparable to that at the exit channel. When these exit and
entrance channel entropies are comparable the number of
available states are also comparable and the overall reactive
flux is influenced by both transition states.
A more detailed view of the potential along the exit
channel is provided in Figs. 2-4. In Fig. 2 a contour diagram
of the potential as a function of
rHF
and
rLiF
(in & with the
LiFH bending angle %i, fixed at its equilibrium barrier top
value of 73.3” is given. In Figs. 3 and 4 contour diagrams of
the potential as a function of first
r
‘r and +,, with ruF
fixed at its barrier top value of 1.207
x , and then of
rHF
and
e,
iFH with rLir fixed at its barrier top value of 1.624 A. These
diagrams suggest that the HF distance provides a good first
approximation to the reaction coordinate in the vicinity of
the primary barrier.
Ill. QUANTUM SCATTERING THEORY
The three-dimensional quantum reactive scattering is
formulated using the coupled-channel adiabatically adjusting
TABLE I. Stationary point structures and energies.
1.9
1.8
1.1 G-F
1.6
1.5
1.4
1 1.5 2 2.5
H-F
FIG. 2. Contour plot of the potential energy surface for the exit channel with
the bending angle LiPH fixed at 73.3”. The energy range is 0.0576-0.927
eV, and the energy increment is 0.0435 eV.
principal axis hyperspherical method.14 The scattering Schro-
dinger equation is solved by expanding the global wave
function in terms of a product of Wigner rotational functions
times surface functions of the two hyperangles 8 and x. After
averaging over the five hyperangles, the resulting coupled
differential equations in the hyper-radius
p
are integrated by
propagating the solution through the different sectors using
the log-derivative method.15 To compute the surface func-
tions at fixed values of
p,
the analytical basis set method16
was adopted for the whole
p
range. After propagation to the
last sector
p
value, asymptotic boundary conditions are im-
posed to determine the scattering matrix element and reac-
tion probabilities. Convergence to within a few percent of the
reaction probabilities and CRP’s with respect to number of
coupled channels has been verified for energies less than 0.5
eV and increasing to 6%-8% for energies greater than 0.5
eV.
IV. VARIATIONAL TRANSITION STATE THEORY
APPROACH
As discussed above, both the entrance and exit channels
are likely to provide important bottlenecks in the reactive
flux. These bottlenecks are expected to act in series to pro-
vide an effective reduction in the reactive flux. If one as-
sumes that the probabilities for passing through these bottle-
Reactants Complex Primary Second Second
barrier well barrier Products
rJs (A) . . . 1.892 1.624 1.562 1.566 1.564
rHF 6)
0.917 0.919 1.207 1.651 2.376 . . .
%iFH . . . 107.3 73.3 69.8 96.8 . . .
E W) 0.000 -0.302 0.182 0.113 0.176 0.157
E+E, (eV) 0.255 . . . 0.303 0.217 0.229b 0.214
%a is the zero point energy from WKB analyses at these stationary points.
bAt the second barrier the zero point energy for the bending motion is neglected due to its highly anharmonic
hindered rotor type nature.
J. Chem. Phys., Vol. 100, No. 7, 1 April 1994
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Yang et al.: Comparison of TST and QST for Li+HF 4919
150
t
100
50
heta
1.4 1.5 1.6 1.7 1.8 1.9
Li-F
FIG.
3. Contour plot of the potential energy surface in the vicinity of the
primary barrier with the HF distance fixed at 1.207 A. The energy range is
0.182-2.055 eV, and the energy step is 0.0624 eV.
necks are in each instance determined by statistics then this
effective flux is directly related to an effective number of
states, given byI
1 1 1 1
N
TE =Nz +a -%-ii’
(1)
In Eq. (1) Nlnt is the minimum in the number of available
states for the entrance channel and NLXi, is the corresponding
quantity for the exit channel. Meanwhile, Nwell is the maxi-
mum in the number of available states in the region separat-
ing the bottlenecks. Here, we will make the assumption that
N well
is large and focus on the entrance and exit channel
numbers of states. This leads to the result
(2)
which is equivalent to that obtained by Light18 on the basis
of strong coupling assumptions in the complex region.
Within statistical assumptions the number of states represents
the CRP and thus N,, as determined from Eq. (2) provides
our approximation to the results of the scattering theory cal-
culations.
150
theta
100
50
1 1.5
2 2.5 3
H-F
FIG. 4. As in Fig, 2 but with the LiF distance fixed at 1.624 %, and &,
varied. The energy range is 0.0287-1.278
eV,
and the energy increment is
0.042 CV.
A..The entrance channel, Li+HF
The formation of the LiFH complex from the Li and HF
reactants proceeds along a barrierless reaction path. We be-
gin by focusing our attention on large separation distances
where the two reactants are only weakly interacting. The
reaction coordinate is then well approximated as the distance
between the Li atom and the HF center-of-mass. The vibra-
tional and rotational states of the HF fragment and the orbital
angular momentum describing the motion of the Li***HF
line-of-centers are also then all approximately conserved
quantities. Furthermore, the HF vibrational and rotational en-
ergies are each nearly independent of reaction coordinate at
these large separations and can thus be taken as constants
equal to their reactant values. The sum of the potential and
the orbital kinetic energy provides an effective potential
which depends on the value of the orbital angular momentum
quantum number 1. Denoting the energies at the maximum in
the effective potentials as
Ey
and assuming that a state is
reactive if and only if there is enough translational energy to
pass over this effective barrier suggests that for total angular
momentum J=O the number of available states may be writ-
ten as
N,&E,J=O)= c O(E-E;"=;HF-E:;F-E~F),
%Fg/HF
(3)
where 0 is the Heaviside step function. This procedure for
evaluating the number of available states has been termed
phase space theory (PST).19
When the two reactants get closer together, this picture
breaks down in a variety of ways. The two fragments begin
to interact quite strongly with one another both in attractive
and repulsive fashions depending on their relative orienta-
tion. As these interactions increase in strength, the orbital
motion, and the HF rotational and vibrational motions all
become mixed together with a gradually increasing loss of
adiabaticity. The assumption of a center-of-mass separation
distance for the reaction coordinate also begins to break
down. The distribution of energy amongst the different de-
grees of freedom becomes more and more random. Eventu-
ally a statistical rather than adiabatic description of the reac-
tion becomes most appropriate. Most importantly, if the
bending motions become repulsive enough, the number of
available states at close separation can become significantly
smaller than the PST number.
Here, we apply a recently developed variational statisti-
cal procedure” for evaluating the number of available states
for the present entrance channel at the shorter separations.
This approach is particularly appropriate for cases where the
transition state is at such a close separation that a statistical
rather than adiabatic description for the bending and rota-
tional degrees of freedom is most appropriate. The basis of
this procedure, as it applies to the present reaction, is an
assumed separation of the HF vibrational motion from the
remaining degrees of freedom. The HF vibrational degree of
freedom is then treated quantum mechanically with its en-
ergy levels determined via a reaction path dependent WKB
analysis. The remaining rotational and relative rotational
J. Chem. Phys., Vol. 100, No. 7, 1 April 1994
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modes are then treated via Monte Carlo evaluation of classi-
cal phase space integrals. The low frequency nature of these
motions suggests that a classical treatment is satisfactory as
has been demonstrated for other related reactions.*l The de-
tails of this approach as it applies to a triatomic reaction such
as the present one are provided in Ref. 22.
An important aspect of this approach is its variational
treatment of the transition state dividing surface. As is well
known, transition state theory type treatments depend implic-
itly on the choice of the dividing
surface
with the value
obtained for any one given dividing surface providing an
upper bound to the rate constant within a classical frame-
work. In variational transition state theory, the dividing sur-
face is varied to find the surface which provides the lowest
and therefore best estimate for the reactive flux. In typical
implementations a reaction coordinate is defined (either ex-
plicitly or implicitly as in reaction path type approaches) and
the value of this reaction coordinate is varied to determine
the minimum estimate. In the approach described in Ref. 20,
and employed here, both the
definition
of the reaction coor-
dinate and its
value
are explicitly varied. In particular, the
transition state dividing surface is defined in terms of a fixed
distance between two points; one point related to each of the
two fragments. Here, these points are taken to be the Li atom
and a point somewhere along the HF axis. A minimization of
the calculated number of states with respect to both the dis-
tance between these points and the location of the point
along the HF axis provides our best estimate of NzJE,J).
In the present calculations, this minimization was per-
formed on a 0.2 A grid in the point to point distance and a
0.2 8, grid in the location of the point along the HF axis.
.2X 10’ sampling points were used in each of the Monte Carlo
integrations providing error bars of about 5% or less. This
procedure was repeated for a range of energies and then fit to
an analytical function to provide an estimate of NL”,(E,J
= 0) for any arbitrary
E.
For comparative purposes with the
scattering theory calculations, these statistical calculations
were then partially quantized according to the ad hoc proce-
dure described in Ref. 22. A fully quantized number of states
were then obtained as the greater integer less than this par-
tially quantized result. This approximate quantization brings
our classical calculation of N,,,(R;E,J=O) at large separa-
tions
R
into semiquantitative agreement with quantum PST
at low energies. ‘
We have also performed phase space theory calculations
as outlined at the beginning of this section with one excep-
tion; the long range effective barriers were deemed negli-
gible since we are considering only very small values of 1.
The results of the PST and the variational TST calculations
are presented in Table II. These results indicate that the
variational estimates of Nznt for the inner transition state be-
come increasingly smaller than the PST estimate as E in-
creases from 0.30 to 0.60 eV. Correspondingly, the bottle-
neck in .the reactive flux for the entrance channel occurs at
separations of about 2.4-3.0 8, for energies exceeding about
0.30 eV. The optimized value of 0.0 A for
Rf,
, the distance
of the HF axis fixed point from the F atom, suggests that a
LiF bond distance provides a good approximation to the re-
action coor’dinate in the region of the inner transition state.
TABLE II. Number of available states for the entrance channel.
Energy Nt,,,, N~WTXT
(eV)
(E,J=O)' (R',R;,;E,J = O)b R+(A) R:, (hd
0.298 4 4 3.0 0.0
0.341 6 5 2.6 0.0
0.385 7 6 2.4 0.0
0.428 8 7 2.4 0.0
0.471 9 7 2.6 0.0
0.515 10 8 2.4 0.0
0.558 11 9 2.6 0.0
0.602 12 9 2.4 0.0
4920 Yang et al.: Comparison of TST and GIST for Li+HF
‘Number of available states for the entrance channel as determined by PST.
bNumber of available states for the entrance channel at the inner transition
state as determined by variational transition state theory.
CBond length between Li and F at the variationally determined minimum in
N .
“Dziance from HF fixed point to F at the variationally determined minimum
in N,,, .
For energies below 0.3 eV the PST result is expected to
be lower than the variational TST result. Thus, the value for
NL,JE,Y) was taken to be given by PST for energies below
0.3 eV and then switched to the variational TST result for the
inner transition state region at higher energies. In previous
studiesz2 of the reactions of He and Ne with Hl we. have
discussed the possible effect of the presence of a maximum
in the reactive flux between two separate transition states for
the barrierless channel alone (i.e., an outer PST type transi-
tion state at separations of about 10 A and an inner transition
state at separations of about 3 A with a maximum in the flux
in between). Here, there was not found to be any substantial
maximum in the flux between the two transition state re-
gions. As a result, any effect due to the occurrence of two
separate transition states is expected to be negligible [e.g.,
consider Eq. (1) in the limit where Nwell is approximately
equal to either Nz”, or Nap,] and was not further considered
here.
Interestingly, the surface functions determined in the
scattering theory calculations” provide an alternative route
for estimating the number of available states for the entrance
channel. That is, the number of available states at the en-
trance channel transition state is approximately given by the
number of surface functions which correlate with reactant
channels and whose adiabatic energy maxima are below the
given energy. This is only an approximate relation since
there are kinetic couplings between the hyperspherical dis-
tance and angles. Furthermore, there is no optimization of
the form of the reaction coordinate in this surface function
based approach. A comparison of these surface function en-
ergy maxima with the variational transition state theory esti-
mates of the energetic locations of the steps in the CRP’s is
presented in Table III. Interestingly, both inner and outer
maxima arise in these adiabatic surface function energies.
For example, see Fig. 1 of Ref. 12 where the inner maxima
are seen to occur at
p
values of about 8 bohr and the outer
maxima occur in the asymptotic region at p values of greater
than 30 bohr. At low energies the outer maximum dominates
and good agreement is obtained with the PST estimates im-
plicit in the present variational statistical approach, At ener-
J. Chem. Phys., Vol. 100, No. 7, 1 April 1994
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TABLE III. Energetic thresholds for the entrance channel.
,@F
umcr ES@
D”t-2, ET
State
i (eV) (eV)
iW
1
0.213 0.256 0.255
2 0.225 0.261 0.260
3 0.239 0.271 0.271
4 0.267 0.287 0.286
5 0.292 0.307 0.304
6 0.332 0.334 0.353
7 0.379 0.368 0.413
8 0.434 0.398 0.476
9 0.497 0.440 0.540
VIC energy corresponding to the inner maximum in the effective barrier for
the ith state obtained from the analysis of the surface function energies.
bThe energy wrresponding to the outer maximum in the effective barrier for
the ith state obtained from the analysis of the surface function energies.
The energy corresponding to the location of the
ith step in
the
present
“quantized” variational transition state theory calculations of the number of
available states.
gies above 0.34 eV the inner maximum dominates and quali-
tative agreement is found with the inner transition state
estimate from the variational calculations (e.g., the numbers
are never greater than one different). When these surface
function energies are available, as in the present reaction, the
surface function based procedure is considerably easier to
implement than the variational TST approach. However, the
variational TST approach has a much wider range of appli-
cability (i.e., it may be applied to essentially any barrierless
reaction) and includes an explicit optimization of the form of
the reaction coordinate. Thus the focus of the current article
will be in testing its validity rather than that of the surface
function based approach.
B. The exit channel, LiF+H
In our analysis of the transition state for the exit channel,
we focus our attention on the primary barrier which is the
first stationary point along the reaction path after the LiFH
complex. The secondary barrier, although of nearly equiva-
lent height, is not expected to provide an important bottle-
neck for the reaction since the bending potential is much
weaker in this region (e.g., see Fig. 4). That is, this much
weaker bending potential leads to much more closely spaced
bending energy levels and therefore a greatly increased num-
ber of available states as verified in sample calculations. The
effect of this secondary barrier is thus neglected in the
present calculations.
A standard simple procedure for estimating the number
of available states at the transition state in the presence of a
well defined barrier involves a harmonic normal mode analy-
sis at the barrier top. A more detailed reaction path analysis
has been performed by Dunning and co-workers= employing
a different potential energy surface and provides a good
qualitative description of the dynamics near this barrier top.
In the present case, the harmonic analysis (using standard GF
matrix techniques24) at the primary barrier top yielded vibra-
tional frequencies of 805, and 1182 cm-‘, and an imaginary
reaction coordinate frequency of 915 cm-‘. The two real
frequency normal modes correspond primarily to the LiF
stretching and LiFH bending motions while the reaction co-
ordinate corresponds primarily to a combination of HF
stretching and LiF shrinking motions.
While a harmonic analysis at the barrier top provides a
good first estimate of the number of available states at the
transition state, there are a number of factors which can con-
siderably change this estimate. For instance, as the HF sepa-
ration distance decreases the bending vibrational frequency
may increase. Variational considerations would then suggest
that the transition state occurs at an HF separation slightly
smaller than that at the barrier top. In considering this pos-
sibility, we performed internal coordinate harmonic analyses
for a range of HF separations. These calculations suggested
that the variation of the transition state away from the barrier
top is negligible-at least for the energies of interest here.
The reaction path analysis of Miller and co-workers25 would
provide a more meaningful estimate of these effects but was
deemed beyond the scope of the current work.
Another important consideration involves the effects of
anharmonicities on the vibrational frequencies at the barrier
top. The simplest procedure for estimating the effect of these
anharmonicities is to perform a WKE? analysis within each of
the real vibrational normal modes at the barrier top. Once the
quantized vibrational energy levels on the first barrier top are
resolved, the number of available states may then be evalu-
ated as
(4)
where
E
is the excess energy, u *, u2 are the two vibrational
quantum numbers, and
E, , ,+ = E, , + Ev2.
An initial implementation of this WKB analysis employ-
ing an internal coordinate (rHF,rLjF, and &i,) based repre-
sentation of the normal modes resulted in certain irregulari-
ties. In particular, the bending normal coordinate includes so
much FH stretching motion that the corresponding potential
reaches regions of very weak bending potential for larger
values of the coordinate. As a result, the potential initially
rises, then declines after reaching a height of 0.50 eV, and
then rises once again, as indicated in Fig. 5. WKB analyses
then become difficult and are likely even physically incor-
rect.
These irregularities in the normal mode WKB analysis
suggest that the optimum transition state surface should in-
stead include some curvature so as to allow the bending po-
tential to remain repulsive. For example, if the bending nor-
mal coordinate was purely the bending angle then the
potential plotted in Fig. 5 would be a smooth monotonically
rising function (e.g., consider Fig. 3). Unfortunately, a quan-
tum mechanically based treatment of such a curvature in the
transition state dividing surface is not easily implemented. A
perturbatively based procedure for partially incorporating the
effect of such curvature has recently been developed by
Miller and co-workers.26
However, such calculations were
again beyond the scope of the present work.
Instead, we have considered a representation of the nor-
mal modes in terms of the Jacobi coordinates for
the reac-
tants. These coordinates are given by the Li to HF center-of-
Yang et a/.: Comparison of TST and QST for Li+HF 4921
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kO.4
ki
8 0.3
0.2 t
0.1’ ’ ’ ’ ’ ’ . ’ . ’ ’
-1 0 1 2 3 4
Normal Coordinate ( i
gy2>
FIG. 5. The potential along the primarily bending normal coordinate at the
primary barrier top. The unit of the normal coordinate is a mass weighted
normal coordinate.
mass separation distance
R,
the HF bond length r and the
angle 0 between the HF bond vector and the vector connect-
ing the Li atom to the HF center-of-mass. These Jacobi co-
ordinates are nearly identical to the internal coordinates due
to the much greater mass of F relative to H. However, there
are no kinetic coupling terms in these coordinates which sug-
gests that one-dimensional WKB analyses in these coordi-
nates might provide a reasonable estimate of potential anhar-
monicity effects. Here, an ad hoc estimate for the quantized
vibrational energy levels which takes into account (i) the
expected accuracy of the harmonic normal mode treatment at
low energies and (ii) the internal coordinate anharmonicity
estimates has been implemented as
E,~ _Euh:rmonic,normal+E~KB,Jacobi_E~:rmonic,Jacobi . (5)
Before discussing the results of the above-described Ja-
cobi coordinate WKF3 analysis it is interesting to also con-
sider how a change of underlying coordinates might affect
the normal coordinate based WKB analysis. In fact, here, the
change of underlying representations for the normal modes
was found to already lead to an improved representation of
the anharmonicity effects. In particular, the one-dimensional
WKB analysis employing the Jacobi coordinate based nor-
mal coordinates did not lead to any irregularities for the
“bending” normal coordinate. This possibility arises due to
the nonlinear relation between the internal and Jacobi coor-
dinates.
The results of the one-dimensional WKB analyses em-
ploying (i) the normal coordinates expanded in the Jacobi
coordinates and (ii) simply the Jacobi coordinates
(R,B)
themselves are given in Tables IV and V. The vibrational
energy levels obtained from Eq. (5) for the Jacobi coordinate
analysis are found to deviate from the normal mode based
energies by less than 0.004 eV for all levels below a total
energy of 0.6 eV. The results presented below employ the
energies obtained from the normal mode based WKB analy-
ses.
A final important consideration involves the effect of
tunneling through the barrier. The primary effect of tunneling
4922 Yang et a/.: Comparison of TST and GIST for Li+HF
TABLE IV. Vibrational energy levels for the stretching mode.8
Vibrational Harmonic WKE3
level normal Jacobi Normal Jacobi
v=o 0.050 0.045 0.051 0.045
v=l 0.150 0.134 0.146 0.130
v=2 0.250 0.223 0.239 0.212
v=3 0.349 0.313 0.328 0.290
v=4 0.450 0.402 0.413 0.364
“Energies are in eV and are with respect to the primary barrier’s threshold.
is to smooth out the step function behavior in N~~,(E,J
= 0). The relatively high frequency for the reaction coordi-
nate motion suggests that this may be important for the
present reaction. The simplest procedure for estimating the
effect of tunneling involves an assumption of a separable
quadratic barrier along the reaction coordinate. With this as-
sumption the effect of tunneling leads to an expression for
N~ait(E,J = 0) given by27
NQE7J=O)= C P(E-Eu1,u2)7
Vl
.v2
where
(6)
P(E) =
exp(
2rElh tib)
1 +exp(2rrE/htib) ’
(7)
and o, is the magnitude of the imaginary frequency at the
barrier top. Alternative more accurate procedures for estimat-
ing the effect of tunneling have been developed (see, e.g.,
Ref. 9). However, the present treatment is expected to pro-
vide at least a qualitative indication of the smoothing effect
of tunneling on the CRP and suffices for the present purposes
of investigating the validity of TST.
A plot of the values of
Nd,i,(E,J
= 0) calculated within
(i) the harmonic approximation without tunneling, (ii) the
anharmonic approximation without tunneling, and (iii) the
anharmonic approximation with tunneling is provided in Fig.
6. From the results presented there, the anharmonicities are
seen to provide a minor increase in the number of available
states.
Meanwhile, the tunneling correction almost com-
pletely removes any indication of the step function increases.
V. COMPARISON BETWEEN SCAlTERlNG THEORY
AND TRANSITION STATE THEORY
In Fig. 7 the transition state theory estimates of the num-
TABLE V. Vibrational energy levels for the bending mode.
Vibrational Harmonic WKB
level normal Jacobi Normal Jacobi
v=o 0.073 0.062 0.072 0.061
v=l 0.220 0.187 0.210 0.176
v=2 0.367 0.311 0.340 0.281
u=3 0.513 0.435 0.461 0.375
v=4 0.660 0.569 0.575 0.460
v=5 0.806 0.684 0.681 0.534
‘Energies are in eV and are with respect to the primary barrier’s threshold.
J. Chem. Phys., Vol. 100, No. 7, 1 April 1994
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10 . ,
8- ,’
6-
4-
2-
O-
I * I I I t
0.3 0.4 0.5 0.6
Energy (ev>
FIG. 6. The exit channel number of available states NLX,(E,J = 0) vs
energy at various levels of approximations. The dotted line corresponds to
the harmonic result with no tunneling corrections, the solid line corresponds
to the anharmonic result with no tunneling corrections, while the dashed line
corresponds to the anharmonic and tunneling corrected estimate.
ber of available states for the entrance channel, the exit chan-
nel, and the effective number calculated according to Eq. (2)
are plotted vs energy for energies ranging from threshold up
to 0.60 eV. The exit channel is seen to provide the dominant
transition state for all energies considered. However, the en-
trance and exit channel numbers become closer and closer as
the energy rises. This decreasing difference between the en-
trance and exit channel number of states leads to an increas-
ing divergence of the effective number of states from the exit
channel value.
Also plotted in Fig. 7 are the accurate values for the CRP
calculated via three-dimensional scattering theory. These
CRP’s have been averaged over a 0.01 eV range with a
10
8
2
0
I . I I , I I
0.3 0.4 0.5 0.6
Energy ( eV 1
FIG. 7. The CRP and
the
statistical contributions to it vs energy. The solid
line corresponds to the three-dimensional quantum scattering calculation.
The dotted line corresponds to the minimum in the number of available
states for the entrance channel as calculated by variational transition state
theory. The dashed line corresponds to the anharmonic and tunneling cor-
rected number of available states for the exit channel. The dash-dotted line
corresponds
to
the effective number of available states calculated
from Eq. (2).
3.0 -
2.5 -
1.5 -
1.0 -
0.5
,*-a’
- ,’
-I
-’ 0.0 - , I I
0.3 0.4 0.5 0.6
Energy (ev>
Yang et a/.: Comparison of TST and QST for Li+HF 4923
FIG. 8. Three-dimensional quantum scattering calculations of the CRP. The
dashed line is the raw data that has been broadened with a Gaussian of
FWHM=O.Ol eV. The solid line is the raw data and is offset by +l for
clarity.
Gaussian weight.28 The effective number of states from the
transition state theory calculations are seen to provide a good
approximation to these averaged CRP’s for the full energy
range-differing by at most about 20%. The generally good
agreement observed here suggests that the entrance and exit
channels are indeed providing an effective reduction in the
flux and that statistical estimates are providing a good esti-
mate of the flux through each of these channels. The minor
overestimate at high energies may be an indication that the
present analysis slightly overestimates the number of states
at the barrier top for the exit channel.
In Fig. 8, the unaveraged (raw data) quantum scattering
CRP’s are shown along with the Gaussian-averaged results.
The fine structure in the raw data is due to quantum dynami-
cal resonances. The density of resonances is greatest near
threshold (between 0.26 and 0.35 eV). Here, the resonances
have been attributed’* to hindered rotational motion of the
Li+FH. Above the energy of 0.35 eV, the density of reso-
nances is much reduced.
VI. CONCLUDING REMARKS
The present transition state theory type calculations have
provided a variety of results. First, the importance of short
range interactions in leading to an inner transition state re-
gion for the entrance channel has been demonstrated. The
importance of anharmonicity and tunneling effects for the
exit channel have also been demonstrated. Furthermore, they
have indicated the importance of considering the combined
effect of both the entrance and exit channel transition state.
The generally good agreement obtained between the statisti-
cal and three-dimensional quantum scattering theory calcula-
tions suggests that the various assumptions present in the
statistical calculations are likely appropriate; particularly for
energies below about 0.54 eV.
J. Chem. Phys., Vol. 100, No. 7, 1 April 1994
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4924 Yang et
a/.:
Comparison of TST and QST for Li+HF
ACKNOWLEDGMENTS
This research was supported in part by NSF Grant No.
CHE-9215194 and performed in part under the auspices of
the U. S. Department of Energy under the Laboratory Di-
rected Research and Development Program.
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