J. Phys. Chem. 1983, 87, 1255-1263
diameters given in Table VI. A further similarity is that improved fits were obtained with the MSA model if the ionic size was temperature dependent. It is clear that the hard-core diameters of Table VI (or the ionic sizes from the MSA model) are physically realistic only if they refer to the solvated ions. In this respect they are consistent with generally accepted trends in hydration. In conchsion we mention additional experimental results which would be highly desirable. These include enthalpies of dilution for KCl(aq) and CsCl(aq) above 200 “C and the same for LiCl(aq) above 100 “C, heat capacity measurements for all three electrolytes above 130 “C (existing data” are not in very good agreement with other thermodynamic results), and, in an interesting region, the
1255
heat capacities of LiCl(aq) and CsCl(aq) below 25 “C.
Acknowledgment, This research was sponsored by the Division of Chemical Sciences, Office of Basic Energy Sciences of the U.S. Department of Energy under Contract W-7405-eng-26 with the Union Carbide Corporation. Registry No. LiCl, 7447-41-8; KC1,7447-40-7; CsCl, 7647-17-8.
Supplementary Material Available: Tables VII-XVIII (24 pages) contain osmotic and activity coefficients, apparent molal enthalpies, and excess heat capacities at rounded molalities and temperatures for LiCl(aq),KCl(aq), and CsCl(aq). Ordering information is given on any current masthead page.
-+
Theoretical Analysis of the Quantum Contributions to the Reactions H,(v = 1) H H 2 ( v ’ = 0, 1) and H,(v = 1) D H H D ( v ’ = 0, 1)
+
+
+H
-
Robert B. Walker’ Theoretical Dlvision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
and Edward
F. Hayes
Dlvlslon of Chemistty,National Science FoundaNon, Washlngton, D.C. 20550 (Received: August 20, 1982)
Detailed quantum-dynamical calculations on the Siegbahn-Liu-TruhlaHorowitz (SLTH) surface are reported for a rotating linear model (RLM)approximation with and without corrections for bending zero-point energy. These dynamical results predict that there are substantialprethreshold quantum contributions to state-selected cross sections and rate constants for both of the title reactions. However, the mechanisms for these prethreshold quantum effects are not the same. For Hz(v = 1) + H, a threshold resonance is responsible for the large prethreshold quantum contribution: 65% of the total rate of 300 K. For Hz(v = 1) + D, tunneling is found to be large, leading to a 66% prethreshold quantum contribution at 300 K. These large quantum corrections are not large enough to provide an explanation for the previously identified discrepancy between the experimental and classical theoretical rate constants for these reactions.
I. Introduction The main purpose of this paper is to examine the quantum contributions to the reactions H + H2(v = 1) Hz(u’= 0, 1) + H and D + Hz(v = 0) HD(v’ = 0, 1) + H. We became interested in this reaction after reading the Mayne and Toennies paper’ in which they identified a major discrepancy between the experimentally determined2“ and theoretically predicted rate constants. The theoretical predictions were derived from quasi-classical trajectory studies using the Siegbahn-Liu-Truhlar-Horowitz (SLTH) potential energy s ~ r f a c e . ~In J view of the excellent agreement between experiment and theory for
-
-
(1)H.R.Mayne and J. P. Toennies, J . Chem. Phys., 75,1794 (1981). (2)R. F. Heidner, 111, and J. V. V. Kasper, Chem. Phys. Lett., 15,179 (1972). - .. -,. (3) Y. M. Gershenzon and V. B. Rozenshtein, Dokl. Phys. Chem. (Engl. Transl.),221,664 (1975). (4)E. B.Gordon, B. I. Ivanov, A. P. Perminov, V. E. Balalaev, A. N. Ponomarev, and V. V. Filatov, Chem. Phys. Lett., 58,425 (1978). (5) M. Kneba, U. Wellhausen, and J. Wolfrum, Ber. Bunsenges. Phys. Chem., 83,940(1979). (6)P.Siegbahn and B. Liu, J. Chem. Phys., 68, 2457 (1978);B. Liu, ibid.,58, 1925 (1973). (7)D.G.TNhlar and C. J. Horowitz, J . Chem. Phys., 68,2466(1978); 71,1514 (E) (1979).
.
0022-3654/83/2087-1255$01.50/0
the H + Hz(v = 0) reaction, this reported discrepancy between experiment and the identical theoretical approach for the H H2(v = 1) reaction is particularly surprising. A short review of the status of both the theoretical and experimental efforts on this system has been recently given by Schatz.8 Very recently, Glass and Chaturvedig have completed a new measurement of the D + H2(u = 1) reaction rate using a discharge flow apparatus coupled with EPR detection of H atoms produced by the reactions. They obtain a rate constant which is over 1order of magnitude lower than that measured previ~usly.~ These new measurements substantially reduce the discrepancy between theory and experiment for the D Hz(u = 1) reaction and, by implication, cast doubt on the accuracy of the measurements of the H + H2(v = 1)rate. A possible contributing factor to the lack of agreement between the experimental and theoretical rate constants is that quantum-dynamical effects not included in the
+
+
(8)G. C . Schatz in “Potential Energy Surfaces and Dynamics Calculations”, D. G. Truhlar, Ed., Plenum Press, New York, 1981,p 287-310. (9)G.P. Glass and B. K. Chaturvedi, J. Chem. Phys., 78,3478(1982).
0 1983 American Chemical Society
1258
The Journal of Physical Chemistry, Vol. 87, No. 7, 1983
Walker and Hayes
lz
30
0
0 d
20
10
Flgure 2. Coordinates for rotating linear model. Flgure 1. Contour plot for the SLTH potential energy surface for collinear H-H-H configurations. The contours are spaced every 0.2 eV with zero taken as the minimum-energy point in the entrance channel. The X marks the approximate location of the H,(v = 1) vibrationally adiabatic barrier.
quasi-classicaltrajectory studies are more important than simple tunneling corrections would suggest. A possible source of such large quantum contributions might be the existence of preclassical threshold resonances, such as were discussed recently for the reaction F + Hz(u = 0) HF(u’ = 2) + H.l“-12 In order to test this possibility and to gain better insight into the dynamics of the H3 system, we have performed exact quantum-dynamical calculations for the Hz(u = 1) within the rotating linear apreaction H proximation using the SLTH surface. The rotating linear model has been used previously by Child,13by Wyatt,’* and by Connor and Child15for approximate dynamical studies. To our knowledge this is the first time that exact quantum scattering results have been obtained for this model. The methods used here are similar in spirit to the recent work of Bowman, Ju, and Lee,16J7where exact collinear quantum reaction probabilities are incorporated into a three-dimensional (3-D) transition-state-theory (TST) description of reactions. In their approach, energy-shifted collinear reactive scattering information is used at each partial wave in a three-dimensional formulation of the collision dynamics. The energy shift applied to the collinear S matrix is defined by the difference in the centrifugal barrier heights evaluated at a transition-state geometry of the collision complex. This approach has been used to predict rate constants for both Hz(u = 1)+ H and Hz(u = 1) D. By comparison, the rotating linear model can be viewed as a dynamical generalization of the TST idea, in which the effects of the centrifugal barrier are included in the numerical solution of the scattering dynamics at each partial wave. In related w ~ r k , ~ this *J~ transition-state-theory approach has been used to model the three-dimensional reactive differential scattering cross sections for the F H, and F + HD reactions. Bowman and Leez0Sz1have also calculated rate constants
-
+
+
+
(10)E. F. Hayes and R. B. Walker, J. Phys. Chem., 86,85 (1981). (11)V. K. Babamov and A. Kuppermann, J. Chem. Phys., 77,1891 (1982). (12)E.Pollak, J. Chem. Phys., 74,5586 (1981). (13)M. S. Child, Mol. Phys. , 12,401 (1967). (14)R. E.Wyatt, J. Chem. Phys., 51,3489 (1969). (15)J. N. L. Connor and M. S. Child, Mol. Phys., 18, 653 (1970). (16)J. M.Bowman, G.-Z. Ju, and K. T. Lee, J. Phys. Chem., 86,2232 (1982). (17)J. M.Bowman, G.-Z. Ju, and K. T. Lee, J. Chem. Phys., 75,5199 (1981). (18)J. M.Bowman, K. T. Lee, and G.-Z. Ju, Chem. Phys. Lett., 86, 384 (1982). (19)K. T.Lee and J. M. Bowman, J.Phys. Chem., 86,2289 (1982). (20)J. M. Bowman and K. T. Lee, Chem. Phys. Lett., 64,291 (1979).
for the H,(u = 1) + H reaction within the quantum sudden approximation. In another recent study Baer, Mayne, Khare, and KouriZ2reported 10s cross sections for the reaction of Hz(u = 1) with H. Each of these quantumdynamical studies predicts rate constants that are significantly below the experimental results. 11. Theoretical Approach Representation of the Potential Energy Surface. As indicated in the Introduction, the SLTH surface is used throughout this paper. This surface, which is generally thought to be of chemical accuracy, was developed by Truhlar and Horowitz’ in order to represent the ab initio H3surface calculations of Siegbahn and Liu? In the course of this study the accuracy of this fit was checked by comparing the SLTH functional form with the original ab initio points. All SLTH energy points were found to be within 0.03 eV of the corresponding ab initio points. The fit is clearly excellent. Figure 1 shows an equipotential contour plot for the collinear surface. The coordinate system for this figure is the standard one chosen to diagonalize the kinetic energy, so that
Re
(pH’,HH/pHH)1/4[rHH’ re
+ pHH/mHrHHI
= (k’HH/pH’,HH)1/4rHH
(1)
where rwz is the distance from the reacting hydrogen atom, H’, to the nearest atom in the reactant hydrogen molecule, p H H and pHt,HH are diatomic and atom-diatom reduced masses, and mH is the mass of the hydrogen atom. The subscript cy labels the arrangement channel containing asymptotic A + BC configurations. Rotating Linear Model (RLM). The coordinates for the model are shown in Figure 2. The distances RABand RBC locate the three atoms, A, B, and C on a line. The spherical polar coordinates, 6 and 4, orient this axis relative to a Cartesian frame with an origin at the center of mass of the whole system. Although this model leaves out two degrees of freedom (i.e., those associated with diatomic rotation in the reactant and product channels or, alternatively,with bending motion when all three atoms are close together), the dynamics associated with the linear geometries may not be a bad f i s t approximation because the lowest-energy pathways from reactants to products correspond to collinear paths. The Hamiltonian for this model may be written in terms of natural-collision coordinates instead of the channel coordinates (Re,r,) as follows: (21)J. M. Bowman and K. T. Lee, J. Chem. Phys., 72,5071 (1980). (22)M.Baer, H.R. Mayne, V. Khare, and D. J. Kouri, Chem. Phys. Lett., 72,269 (1980). (23)J. C. Light and R. B. Walker, J. Chem. Phys., 65,4272 (1976).
Quantum Contributions to H,(v = 1)
+ H (or D)
The Journal of Physical Chemistry, Vol. 87, No. 7, 1983
(2) where k2is the square of the angular momentum operator, R is the radius of gyration (R = (R,2 + r,2)lI2),and q is a Jacobian factor (q = 1+ K(U)U,where 1 / is~the local radius of a curvature of the natural-collision-codinate reference curve). The corresponding Schrodinger equation (for an incident molecule in vibrational state uo) (3) (A- E)\k,,(u,v,8,4)= 0
1257
the reactive path and added this potential energy term, v b , to the potential for the RLM, eq 7. This adiabatic treatment of bending motion was first described by M ~ r t e n s e n .In ~ ~this work, we use the approach described by Garrett and Truhlar,26who write an approximate Hamiltonian for the bending motion as
where 1
may be reduced to a set of differential equations in the usual way by invoking a partial wave expansion of the total wave function The masses mA,mB, and mc and internuclear distances
m
Qu0(u,v,44)=
c R-'$L,(u,v) YP(8,4)
(4)
1=0
The resulting set of differential equations for
(Pa+ V(u,v) - E)$Lo(U,V)
=0
$ko
is
(5)
where
RAB(u),RBC(U),and RAC(U) correspond to the triatomic arrangement, ABC, shown in Figure 2. The internuclear distances are evaluated on the reaction path determined at each I value. For a fixed position on the reaction path (i.e., fixed u),the bending potential may be approximated as vbe,d(u,8)
The index 1 in eq 4 is the total angular momentum quantum number for the collision system and is equivalent to the L index used by Child,13J5the 1 used by Wyatt,14 and ~ ~ ~cou~ the J used by Bowman and ~ o - w o r k e r s .The pled-channel equations obtained from eq 6 are similar to those encountered in collinear reactive scattering calcul a t i o n ~ . However, ~~ in the present case the boundary conditions are different and the effective potential depends on 1
VI = V(u,v)
h2 + -(l(Z 2pR2
+1)
+ 1)
(7)
The working set of coupled equations is obtained by expandin $' (up)as a product of translational wave functions, u and vibrational wave functions $,(u,v)
tN(,\
N
$;,(UP) =
c ft&)
u=l
4u(u,v)
(8)
The boundary conditions for f'u,(u) are15 fLou(u)=u
St',;'
exp[ikv0u t i ( l + 11771 -t exp(-ik,u) UZ ( h , o / k , ) l ~ z exp(ih,u) S~~~ (9)
where St(2 is a complex reflection coefficient, St(,"is a complex transmission coefficient, and k,, and k, are the usual asymptotic incoming and outgoing wave vectors. The unknown translational wave functions, f'uo,(u)and the scattering matrices may be obtained by making only minor modifications in the scattering code RXNlD.% Bending Correction t o RLM (BCRLM). One of the shortcomings of the RLM approach14is that the effect of the triatomic bending zero-point energy is not included. As a result the reaction thresholds are lower than would be the case for a full 3-D calculation. In order to approximate the effects of bending, we have determined a bending zero-point energy correction at each point along (24) R. B. Walker, QCPE Program No. 352, Department of Chemistry,
Indiana University, Bloomington, IN.
= 1/2Kb(u)e2 + 9b(u)O4
(12)
Our implementation of the bending correction differs slightly from the methods described by Garrett and Truhlar. In our approach, the reaction path is defined by the minimum vibrational energy at each value of the propagation coordinate u,and not by the (unique) minimum-energy path (MEP) defined by a steepest-descent method. Our approach is motivated by numerical convenience and leads to a path (and hence a bending potential) which depends slightly on the choice of the polar origin of the natural-collision-coordinatesystem. We also determine the coefficients Kb and q b in eq 12 by a leastsquares fit to the actual bending potential (sampled every 5 O between 8 = 0" and 8 = 4 5 O ) rather than by evaluating higher derivatives of the potential at 8 = 0. Finally, Garrett and Truhlar26 use a perturbation-variation formula to calculate the eigenvalues of the Hamiltonian in eq 10, whereas we find it more convenient to expand the bending wave function as a linear combination of harmonic oscillator functions with force constant K b . The lowest eigenvalue can be obtained to four decimal places (in eV) using only a few terms in the expansion (i.e., typically five to eight symmetric harmonic oscillator functions). Since a linear triatomic has two degenerate bending modes, the bending zero-point energy correction term, v b , is given by vb(u)
= 2@(u)
(13)
where Eb(u) is the lowest eigenvalue of eq 10 for the specified value of u. In all the work described here, we considered the effects of only the lowest bending state of the collision complex. However, we should note that, in their TST approach, Bowman, Ju, and Lee16suggest that it may be more appropriate to interpret the dynamics so obtained as coming from a rotationally averaged effective state. The equations that we solve here to obtain the I = 0 BCRLM dynamics are effectively the same as the dynamics which Bowman, Ju, and Lee16 label as CEQB. The only difference arises from the additional 1/R2 effective potential term which is present in the RLM equations (see eq 7). We have run (25) E. M. Mortensen, J. Chem. Phys., 48, 4029 (1968). (26) B. C. Garrett and D. G. Truhlar, J.Am. Chem. SOC.,101,4534 (1979).
1258
The Joumal of Physical Chemistry, Vol. 87, No. 7, 1983
Walker and Hayes
-2
2
0
-2
0
2
-
ARC LENGTH (a,)
H2 -b H(v = 0, 1, 2) as a function of position (arc length) Figure 3. RLM vibrational correlation diagram (solid curves) for reaction H i- H, along the minimum-energy path for total angular momentum (a) I = 0, (b) I = 10, ( c ) I = 20. Energy is in eV. The lowest dashed curve is the energy of the minimum-energy path. In the shaded region there is some ambiguity in defining the energy curves (see text).
calculations neglecting this 1/R2term altogether, with no significant effect on either the magnitudes or phases of the RLM S matrix. The BCRLM dynamics for 1 > 0 differs from the 1 = 0 dynamics by the presence of the effective centrifugal potential energy term. Calculation of Cross Sections and Rate Constants. The partial wave cross sections for reactive scattering may be related to the reactive scattering probabi1itiesl3-l5as follows:
where Et is the translational energy for the initial channel. The labels for the reactant vibrational quantum number and the product vibrational quantum number are u and u’, respectively. The reduced mass p is cc = m ~ ( + m m~ c ) / ( m + ~ m~ + mc) (15) The total cross section, avvf(E), is given by
L %d(E) =
c UtVlE)
l=O
(16)
where I, needs to be taken large enough to achieve convergence. For the energy ranges and systems treated in this paper, we have found it satisfactoryto use 20 < 1< 30. The relationship between the state-selected rate constant, kuvl,at temperature T,and the energy-dependent state-selected cross section, avvl,is
(17) where kB is Boltzmann’s constant and N is Avogadro’s number. One simple way to quantify the prethreshold quantum contributions to rate constant is to replace the total cross section by the sum over partial wave cross sections (eq 16) and, for each 1, divide the integrand into a “quantum” energy region corresponding to 0 IEt I Elbarrier and a “classical” region corresponding to Eiarrier5 Et I Following this prescription we have 03.
and k:vl, the classical contribution, is given by kfvf =
The prethreshold quantum fraction of the rate constant, Fvvl,may then be defined as Fuvl= k$$/kvvf (20)
111. Results For each set of calculations it was necessary to ensure that the results were converged with respect to several parameters. In this study we have worked with a primitive harmonic oscillator basis set of modest size (16 functions) and a smaller set of coupled equations based on the local internal eigenstates. The number of coupled equations was generally about 10. We have found this level to be sufficient to accurately reproduce resonance features for several systems.10~27-29 H2(u = I) + H. For each 1-dependent RLM effective potential energy surface, eq 7 , we have determined a set of vibrationally adiabatic correlation curves appropriate to the steepest-descent minimum-energy path (MEP). We used a modified vibrational potential obtained by moving perpendicular to the MEP. For example, if we are on the reactant side of the saddle point and move away from the concave side of the MEP, we eventually fall into the product vibrational well (at an angle). To avoid confusing the correlation diagram with eigenvalues from the product well, we remove the product well (only) by modifying the potential so that it never falls below the value it reaches at the maximum between the two wells. The shaded region in Figure 3 (and in other correlation diagrams to follow) indicates the energy region above this maximum. Only those eigenvalues below the shaded region indicate vibrational states localized in the reactant well. The energy levels of the (modified) vibrational potential are determined by a five-point finite difference method. Two finite difference solutions are then combined by Richardson extrapolation to eliminate the leading fourth-order error term. The resulting vibrational correlation diagrams are plotted in Figure 3, for 1 = 0, 10, and 20. Garrett and Truhlar30have reported similar energy correlation curves for the H3 system on a similar collinear potential energy surface. For 1 = 0, the u = 1 correlation curve has symmetric barriers in the reactant and product regions. For 1 = 10, the symmetric reactant and product barriers are still present but the well between these two barriers is not as deep as in the 1 = 0 case. For 1 = 20, the well has been completely eliminated-only one barrier exists in the symmetric stretch portion of the surface. In Figure 4, the H2(u = 1) vibrationally adiabatic entrance channel barriers are plotted as a function of 1 for ~~~
~~
(27)J. T.Adams, Chen. Phys. Lett., 33, 275 (1975). (28)B. C. Garrett, D. G. Truhlar, R. S. Grev, and R. B. Walker, J. Chem. Phys., 73, 235 (1980). (29)R. B.Walker, Y. Zeiri, and M. Shapiro,J. Chem. Phys., 74,1763 (1981). (30)B.C.Garrett and D. G. Truhlar, J. Phys. Chem., 83,1079(1979); 84,682 (E)(1980).
Quantum Contributions to H2(v = 1)
+ H(or D) I
I
I
The Journal of Physical Chemistry, Vol. 87, No. 7, 1983
1259
A
20
1.2
15 10
-
n
9.
% .a .-
5
L
Q
0 L
-n
0.30
W
0.90
0.60
I .20
1.50
E tot (eV)
.4 B
0.0
I
I
10.
20.
I
0.
1 Figure 4. RLM adiabatic barrier heights (classical thresholds) as a function of total angular momentum, I , for reaction, H2(v = 0, 1) H +.H H2(v’ = 0, 1). Energy is in eV. The upper two curves are for HAv = l), while the lower two are in HAv = 0). The dashed curves include no bending correction, while the soli curve (0)includes bending zero-point energy.
+
+
9.
n A
0.30
0.60
Oa9O
1.20
E tot (eV)
.4 C 0
-a- .2 n
.8
.9
1.0
1.1
Etot
(ev)
1.2 0
-
0.30
oa90
0.60
1.50
1.20
E tot (eV)
0
Figure 5. RLM angular momentum dependent transition probabilities as a function of energy for the reaction H H2(v= 1) H2(v’ = 0) 4- H. The curves are labeled with the angular momentum values, I = 0, 4, 8, 12, and 16. The point on each curve corresponds to the energy position of the vibrationally adiabatic v = 1 barrier. Energy is in eV.
Figure 6. Perspective plot of the RLM transition probabilities for the reaction H2(v= 1) H H H2(v = 0, 1) as a function of energy and angular momentum, 1. Energy is in eV. Each plot changes the observer’s viewpoint slightly.
0 5 I I 20. As we will see in the next section, the I de-
TABLE I: RLM Rate Constants and Quantum Fractions at Several Temperatures for H,(u = 1 ) + Ha
+
pendence of these barriers is important for understanding the quantum contribution to scattering for this system. Converged scattering results have been obtained for the total energy range 0.30 I E I1.70 eV (in steps of 0.01 eV) for all total angular momentum partial waves in the range I = 0-30. From these data, I-dependent reaction probabilities, partial and total integral cross sections, and thermal rate constants can be computed. In Figure 5, the I-dependent reaction probabilities P,,’(E) are plotted as a function of total energy for I = 0, 4,8,12, and 16. The point on each curve indicates the energy corresponding to the value of the I-dependent vibrationally adiabatic u = 1barrier. Note that for each I value there is a substantial prethreshold quantum contribution similar to that of the F + H2 system.1° Garrett and Truhlar,30in their study of quantum effects in collinear reactions, have noted the importance of such a resonance feature for the H2(u = 1) system in a similar potential energy surface. These reaction probabilities have been plotted in perspective in Figure 6, and it can be easily seen that the resonance energy increases smoothly with increasing I (the “resonance ridge” effect described by Wyatt31). In Figure 7, partial integral cross sections aio(E)and &(E) are plotted as a function of I for several energies. (31) M. J. Redmon and R. E. Wyatt, Chem.Phys. Lett., 63,209 (1979); R. E. Wyatt, J. F. McNutt, and M. J. Redmon, Ber. Bunsenges. Phys. Chem., 86,437 (1982).
-+
K
total reactive rate constantb
quantum fractionC
200 300 400 500
6.9 42.1 114 217
0.79 0.59 0.46 0.37
temp,
a Units are 10” cm3 mol-‘ s-’. See eq 2 1 for definition.
k,, + h , , per eq 17.
In each case, the dominant contribution to the total reaction cross section comes from partial waves I > 0. Total reaction cross sections are computed by summing the partial cross sections, and the energy dependence of the total cross sections is used to calculate reaction rate constants by using eq 17. RLM rate constants and their quantum fractions are reported in Table I for several temperatures. In Figure 8, we show vibrationally adiabatic energy correlation curves similar to Figure 3, but for the BCRLM potential energy surface (eq 7 and 13). The I = 0 and I = 10 correlation curves have symmetric barriers and a well as in the RLM case, but the bending zero-point energy increases the height of the barriers by about 0.1 eV. The BCRLM results obtained for the I = 0 reaction probability are presented in Figure 9 for total reactive probabilities from Hz(u = 0) and H2(u = 1). These results are compared (also in Figure 9) with the analogous RLM results and with full 3-D reactive scattering results of
1260
The Journal of Physical Chemistiy, Vol. 87, No. 7, 1983
Walker and Hayes
TABLE 111: BCRLM Rate Constants and Quantum Fractions at Several Temperatures for H,(u = 1) + Da temp,
i -0
ti-
e,,.
L
K
total reactive rate constantb
quantum fraction'
200 300 400 500
1.2 16.8 84.2 24 1
0.90 0.66 0.50 0.39
h , , t k , , per eq 17
a Units are in 1 O ' O cm3 mo1-ls-l. See eq 2 1 for definition.
0
I
l
10
20
TABLE IV: Comparison of the Total Reaction Rate Constantsa from Experiment and Theory for H,(u = 1 ) at 300 K b
l
Total A n g u l a r Momentum 1
system source
H , ( u = 1)
experiment theoryC classical trajectoriesd quantum RLM quantum BCRLM adjusted classicale
+H
H,(u = 1) t
D
311 t 12f
59
7.6 42 1 29.2 21.5
i
18g
t
1.14
10.1 t 1.2
i:
3.3
16.8 29.9 t 3.6
Units are 10''' cm3 mo1-ls-l. h , , + h , , per eq 17. All theoretical results were obtained by using the SLTH surface. Reference 1. e Obtained by dividing classical trajectory result by 1 minus the quantum fraction. f Reference 4. g Reference 9.
I
1
20 Total Angular Momentum 1 Flgwe 7. RLM partial wave cross sections, d,, and d,,, as a function of I for several energies, E . Energies given in eV. (A) H,(v = 1) H H H,(v' = 0). (E) H,(v = 1) H H + H,(v' = 1). 10
0
-+
+
-
+
TABLE 11: BCRLM Rate Constants and Quantum Fractions at Several Temperatures for H,(u = 1 ) t Ha temp, K
total reactive rate constantb
quantum fractionC
200 300 400 500
11.2 29.2 112 32 1
0.98 0.65 0.33 0.21
~
Units are in 1010 cms mol-, s - ' . See eq 2 1 for definition.
h , , + h , , per eq 1 7
Walker, Stechel, and LighP for total angular momentum,
J,equal to zero. Notice that the threshold for BCRLM is in very good agreement with the full 3-D results. This indicates that the bending zero-point energy correction is a reasonable approximation. In Figure 10 the BCRLM angular momentum dependent reactive transition probabilities, plo(E), are plotted as a function of energy for 1 = 0,4,8,12, and 16. As in Figure 5, the point plotted on each curve indicates the position of the vibrationally adiabatic u = 1 energy barrier. As in the RLM case there is a substantial threshold quantum contribution for each total angular momentum, 1. The BCRLM rate constants and quantum fractions are reported in Table I1 for several temperatures. Total Cross Sections. One of the important questions concerning resonances in reactive scattering, and prethreshold resonances in particular, is whether any manifestation of these resonances might be discernible in energy dependence of the total cross section. In order to investigate this point further, we show the energy depen-
dence at the total reaction cross section from H + Hz(u = 1) in Figure 11 and from H H,(u = 0) in Figure 12. Because the H + Hz(u = 1) 1-dependent threshold resonance is present in a large number of partial waves, and each resonance position is slightly shifted in energy (see Figure 6), the summation over all these partial waves leads to a total cross section that changes quite smoothly with energy-so smoothly, in fact, that, if one were confronted only with Figure 11, it would be difficult to identify any contribution due to threshold resonances. The only visible remnant of these u = 1 threshold resonances, in the total reaction cross section curves, is in the energy dependence of the Hz(u = 0) H cross section curve (see the small inflection near energies of 0.7 eV in Figure 12). D + H p The correlation diagrams for the BCRLM potential and the angular momentum dependent transition probabilities ploare presented in Figures 13 and 14, respectively. These two figures should be compared with the H + Hz(u = 1) results, in Figures 8 and 10. There are two important features to notice: (1)Both systems have pronounced barriers and wells for the u = 1 correlation curve; however, for D + Hz the energy barriers in the reactant product channels are unsymmetrical. (2) Only the H + H2(u = 1) system appears to have a threshold resonance. In Table 111, BCRLM rate constants and quantum fractions of the rate constants are reported for several temperatures. The quantum fractions are about the same as those determined for the H + H2(u = 1) reactions (see Table 11).
+
+
IV. Discussion The results reported in the previous section provide a good basis for analyzing the discrepancy which is apparent between the classical theoretical predictions and the experimental measurements for the reactions H ~ ( = u 1) + H H + H2(u = 0, 1) and Hz(u = 1) + D H + HD(u
-
= 0, 1). (32) R. B. Walker, E. B. Stechel, and J. C. Light, J.Chem. Phys., 69, 2922 (1978).
-
Table IV contains a summary of the experimental and theoretical rate constant measurements for both reactions
Quantum Contributions to H,(v = 1)
+ H (or D)
The Journal of phvsical Chemistry, Vol. 87, No. 7, 1983
1261
1 .6
h
I .2
2
v
0.0 0.4
0
-
ARC LENGTH (a,)
+
+
Flgure 8. BCRLM vibrational correlation diagram (solfd curves) for reaction H H, H, H as a function of position (arc length) along the minimum-energy path for total angular momentum (a) I= 0, (b) I= 10, (c) I= 20. Energy is in eV. The lowest dashed curve is the energy of the minimum-energy path. I n the shaded region there is some ambiguity in defining the energy curves (see text). I,
I
0.8 n
+
cu
0
0
20.
0
-
Y
00
a 0.4
0
t
6 0.0
I
1
.
0.4
0.8
bot
1.2
1.6
(4
I
I
10.
I-
1
I
0. la
0.0
-
-
1.2
0.8 Etrans (e")
0.4
-+ ++
Flgure 11. BCRLM total reactive cross sections (alo all a,,) as a function of energy for the reaction H,(v = 1) H H H2(v' = 0, 1, 2). Energy in eV.
+
0
02 CL
n
.4
-8
1.2
bot
(4
No
1.6
-
Flgure 9. Comparison of energy dependence of the I= 0 transition probabilities. Solid and dashed curves are for RLM and BCRLM, respectively. The points are for the full 3-D J = 0 results (initiil rotation j = 0, summed over all final rotational quantum numbers), ref 29. (A) H,(v = 0) H H H,(v' = 0, 1, 2). (B) H,(v = 1) H H H,(v' = 0, 1, 2).
+
-+
20.
c3
Y
+
-+
0
t
g
10.
0.
I
I
I
I
0.0
0.4
0.8
1.2
Etrans (e")
--a.
+
0
1
I
.8
.9
I
I
I
1.0
1.1
1.2
-
Flgure 10. BCRLM angular momentum dependent transition probabiliiies as a function of energy for the reaction H HAv = 1) H2(v' = 0) H. The curves are labeled with the angular momentum values, I= 0,4,8, 12, and 16. The point on each curve corresponds to the energy position of the vibrationally adiabatic v = 1 barrier. Energy in eV.
+
-+ ++
Flgure 12. BCRLM total reactive cross sections (aoo a,, a function of energy for the reaction H,(v = 0) H H 0, 1, 2). Energy in eV.
+
at 300 K. From the results of the previous section, Tables 1-111, it is apparent that the preclassical threshold quantum contributions to the H2(v = 1)+ H and H2(u = 1)+ D reaction rates are significant. Based on the calculated quantum fractions for these rate constants at 300 K, the
aO2) as H2(v' =
prethreshold quantum contributions account for at least 65% of the full RLM and BCRLM rate constants. This suggests that the full classical trajectory results need to be multiplied by a factor of about 2.9 in order to correct for the prethreshold quantum effeds that are not included int the 3-D classical studies. We need to consider a second quantum effect tending to reduce the rate constants. It is well-known that the 3-D classical results do not include the quantum effects associated with bending zero-point energy. Comparison of the RLM and BCRLM results clearly indicates the importance of this quantum effect, which is included in the BCRLM but not the RLM approach. This quantum effect reduces the RLM rate constant by about a factor of 15. The importance of bending zero-point energy in determining the threshold behavior of reactions has been discussed by a large number of authors.14J6J7*33-39
1262
The Journal of phvsical Chemistry, Vol. 87, No. 7, 1983
2
Figure 13. Vibrational correlation diagram for reaction D
.4
Walker and Hayes
-2
+ H,
-
0
I
.8
.9
I
1.0 Etot
I 1.1
I
1.2
(4
-
I
I
0
2
J
DH 4- H. Labeling as in Figure 9. Negative arc length corresponds to reactants.
Figure 14. Angular momentum dependent transition probabilities as a function of energy for reaction D H,(v = 1) DH(v’ = 0) H. Angular momentum values I = 0,4,8, 12, and 16 are indicated in the figure. The point on each curve corresponds to the energy position of the v = 1 vibrationally adiabatic barrier. Energy in eV.
+
I
-2
ARC LENGTH (a,)
t
.-
2
+
If we adjust the classical result for H2(u = 1)+ H by correcting for the predicted quantum fraction, a rate constant of about 22 X 1O1O cm3 mol-l s-l is obtained at 300 K. This is in reasonable agreement with the BCRLM result at the same temperature (see Table IV). However, from these results, we are led to conclude that the quantum effeds analyzed in this paper are not large enough to bring the 3-D classical trajectory results into agreement with the experimental rate constants for the H2(u = 1)+ H reaction. Only by completely neglecting bending motion, as in the quantum RLM results in Table IV, do we obtain a rate constant of comparable magnitude to experiment. Even with bending corrections, not all effects of reactant rotations or full 3-D motions have been treated in this paper. There is always the chance that other quantum effects will be found when the full 3-D scattering calculations are performed, but we doubt that these effects will be large enough to increase the H + H2(u = 1)theoretical rates up to the values determined experimentally. For any theoretical prediction, the accuracy of rate constants depends on the accuracy of the potential energy surface. As was mentioned in section 11, we verified the accuracy of the Truhlar-Horowitz fit7 to the actual ab initio points and found no significant discrepancies. This leads us to ask the obvious question: “How far off would the SLTH surface have to be to bring the theoretical predictions into better agreement with experiment?” (33)G. C. Schatz and A. Kuppermann, Phys. Rev. Lett., 35, 1266 (1975);G. C.Schatz and A. Kupper”, J. Chem.Phys., 66,4668(1976). (34)R. E.Wyatt in ‘State-to-State Chemistry”, P. R. Brooks and E. F. Hayes, Eds.,American Chemical Society, Washington, DC, 1977,ACS Symp. Ser. No. 56,p 185. (35)B. C.Garrett and D. G. Truhlar, J. Phys. Chem.,83,1915 (1979). (36)B. C. Garrett and D. G. Truhlar, PToc. Nutl. Acud. Sci. U.S.A., 76,4755 (1979). (37)B.C. Garrett and D. G. Truhlar, J. Chem.Phys., 72,3460(1980). (38)B. C. Garrett, D. G. Truhlar, R. S. Grev, and A. W. Magnuson, J. Phys. Chem., 84,1730 (1980). (39)E.Pollak and R. E. Wyatt, submitted to J. Chem. Phys.
Basically what would be needed is a shift in the adiabatic energy barriers by about 0.1 eV. This could result from lowering the energy along the minimum-energy path, lowering the vibrational force constant for motion perpendicular to the minimum-energy path, lowering the bending zero-point energy, or a combination of all three effects. New ab initio energy calculations are beyond the scope of this paper, and so we thought it might be helpful to indicate the region of the surface that may need further exploration. A qualitative indication of the location of this region is shown by the X marked on the collinear contour energy diagram, Figure 1. The center of this X corresponds to RAB= 1.43 bohr and RBC = 2.64 bohr and marks the approximate location of the u = 1vibrationally adiabatic (bending-corrected)barrier on the H + H2(u = 1)reaction path. Since this region corresponds to unsymmetrical geometries, it is conceivable that the ab initio calculations of Siegbahn and Liu6 are not as accurate as this portion of the surface as they are for symmetric geometries. However, on the basis of the work reported here, it is not possible to resolve definitively the present discrepancy between experiment and theory. In particular, if the SLTH surface were adjusted in this region in such a way as to raise the H2(u = 1) + H rate constant to agree with experiment: then the adjusted theoretical H2(u = 1)+ D rate constant would surely overshoot the recent measurements of Glass and Chat~rvedi.~ In our view, all potential sources of difficulty, including the experimental results, merit further scrutiny. Up to this point, we have discussed the prethreshold quantum contributions to the H, D + H2(u = 1)reactions without making any attempt to analyze the mechanisms responsible for these prethreshold quantum effects. In what follows, we will present evidence that strongly suggests that, although the quantum fractions of the rate constants are similar (i.e., about 65% for both H and D at 300 K), the mechanisms responsible for these large quantum effects are entirely different. An important clue to these differences is evident in Figure 15. In this figure the BCRLM rate constant integrands, eq 17, are presented for the reaction H, D + H2(u = 1)for a temperature of 300 K. When H is the reactant, there are two important energy regions in the integrand: (1)translational energies below about 0.19 eV, the I = 0 vibrationally adiabatic barrier for u = 1;and (2) translational energies above this barrier. In contrast, when D is the reactant, the integrand has only one energy region that contributes significantly to the rate constant integrand at 300 K. The integrand curves for D + H2(u = 1)look very much like the integrand curves for the reactions D, H + H2(u = 0)-suggesting that the main prethreshold quantum contribution for this reaction comes from tunneling through the u = 1vibrationally adiabatic barrier (at 0.19 eV for I = 0) in the reactant region. Additional evidence for a different threshold mechanism for these two reactions can be obtained from the Argand
Quantum Contributions to H2(v = 1) I
+ H (or D)
I
The Journal of Physical Chemistty, Vol. 87, No. 7, 1983 1
I
A
1.0
-
n v, 0.0
-
-1.0
-
-E
‘ A
I
I .
.
1263
-
I
(Qj1.30
0.0
.2
.4
.a
.6
1.0
I
I
I
I
.6
.a
-
I
I
E trans
A
... I
B
I]il
t
-S
.4
0,
t
0
“li \ 0.0
.2
.4
io
E trans
-
Figure 15. BCRLM rate constant integrands (eq 17, T = 300 K) as a function of translational energy: (A) H H2(v = 1) -,H2(v’ = 0, 1) H (B)D Hdv = 1) HD(v’ = 0, 1) H. The Integrands are normalized to 1.0 at their maximum value. Energy in eV.
+
+
+
-1.0
+
eo
phase diagram for the 1 = 0 reactive matrix element (see Figure 16). The curvature change in the phase of the matrix element with energy, which is characteristic of a resonance,40is present in the case of the H + H,(u = 1) reaction, but not in the case of the D Hz(u = 1) reaction. Although only the 1 = 0 Argand diagrams are shown here, similar results are obtained for all 1 values that contribute significantly to the rate constant at 300 K. For each of these higher 1 values the Argand diagrams show resonance characteristics when the reactant is H, and no resonance characteristics when D is the reactant. The feature of these two systems that is responsible for this difference in mechanism is evident in the vibrationally adiabatic correlation diagrams for u = 1 (see Figures 8 and 13). For H + H,(u = 1) the reactant and product barriers are symmetrical in shape and size, whereas for D Hz(u = 1) the reactant and product barriers are not symmetrical. The product barrier is about 0.1 eV lower than the reactant barrier. As a result, when H is the reactant, for translational energies below the u = 1 adiabatic barrier, the effective potential is similar to the two-barrier transmission problem, which is known to exhibit resonances.1o@ However, when D is the reactant, only one barrier is encountered in the energy region of interest and, importantly, this barrier is almost as easy for the D to tunnel through as the D Hz(u = 0) vibrationallyadiabatic barrier. Qualitatively one might draw this conclusion by comparing the height and thickness of the u = 0 and u = 1 barriers in Figure 13.
+
+
+
(40) A. Kuppermann in ”Potential Energy Surfaces and Dynamics Calculations”, D. G. Truhlar, Ed., Plenum Press, New York, 1981, Chapter 16, p 375.
-
t -1.0
1 .o
0.0
Re@)
q,,
Figure 16. Argand diagrams for transition matrix element for (A) H H2(v = 1) H2(v’= 0) H and (B)D H2(v = 1) HD(v’= 0) H. Energy range is from 0.8 to 1.3 eV with points at E = 0.80, 0.85, ..., 1.30.
+
+
+
+
-+
Quantitatively, the BCRLM calculations confirm thispredicting a quantum fraction of 0.66 for D + H,(u = 1) and 0.86 for D + Hz(u = 0).
Summary Although the RLM and BCRLM results cannot resolve the discrepancy between experiment and theory for the u = 1 rate constants (especially for the H,(u = 1) + H reaction), these results do provide a clear indication of the importance of prethreshold quantum effects for the reactions of Hz(u = 1) with both H and D. The results also predict that the mechanisms for the prethreshold quantum effects in these two systems are not the same. In addition, although the quantum resultg predict no sharp oscillations in the energy dependence of the total reactive cross section, there may be small but discernable inflections in the cross section curves that could easily be overlooked in the absence of other information. Since the angular distributions may contain useful diagnostic information on resonances, we have initiated studies of the angular distribution. We plan to report these results in the near future. Acknowledgment, We acknowledge useful discussions with R. E. Wyatt, D. G. Truhlar, and J. M. Bowman. This work was performed under the auspices of the U.S. Department of Energy. Registry No. Atomic hydrogen, 12385-13-6; hydrogen, 133374-0; atomic deuterium, 16873-17-9.
